THESIS

This is to certify that the
thesis entitled
INTEGRAL-OPERATOR-BASED PERTURBATION
STUDY OF DIELECTRIC WAVEGUIDES

presented by

Terese M. Sipe

has been accepted towards fulfillment
of the requirements for

Masters degreein Electrical
Engineering

 

 

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INTEGRAL—OPERATOR-BASED PERTURBATION

STUDY OF DIELECTRIC WAVEGUIDES

BY

Terese M. Sipe

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE

Department of Electrical Engineering

1981

ABSTRACT

INTEGRAL-OPERATOR-BASED PERTURBATION
STUDY OF DIELECTRIC WAVEGUIDES

By

Terese M. Sipe

.Exact solutions for natural modes supported by dielectric
waveguides exist only for cores of certain cross-section shapes
having specific refractive-index profiles. Numerical or approximate-
analytical methods are required for other relatively-complex core
configurations. A new perturbation analysis for surface-wave modes
guided by heterogeneous, open-boundary dielectric waveguides of
arbitrary cross-section shape, based upon an integral-operator
description of such modes, is studied in this thesis.

Assuming that an exact field solution and waveguide parameters
exist for a relatively-simple "unperturbed" core configuration, a
transcendental equation in parameter y can be obtained for a small
"perturbation" in the core. A closed-form solution can be obtained

fer weakly-perturbed guides by expressing y = 7(0)

(0).

+ Av and expanding
in a Taylor's series about y
The perturbation analysis is applied to the TE modes of the
graded-index slab waveguide and axially-symmetric circular fiber,
and excellent agreement is obtained between these results and those
of a direct moment-method numerical solution to the corresponding

integral equation.

ACKNOWLEDGEMENTS

The author wishes to express her many thanks to her major
professor, Dr. Dennis P. Nyquist, for his help, patience, and

understanding during the course of this research.

Chapter

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

I.

II.

III.

IV.

INTRODUCTION

INTEGRAL-OPERATOR-BASED PERTURBATION ANALYSIS FOR
DIELECTRIC WAVEGUIDES

2.1

2.2

2.3

2.4

2.5

2.6

Equivalent-Current Description for Heterogeneous
Dielectric Waveguides

Volume EFIE for Unknown Waveguide Field Excited
by Impressed Current

TWO-Dimensional EFIE for Natural Surface-Wave
Modes

Specialized EFIE for TE Modes of Rectangular
Boundary Dielectric Waveguides

Specialized EFIE for Axially-Symmetric TE Mbdes
of Circular-Boundary Dielectric Waveguides

Development of Perturbation Equation for TE Modes
of Rectangular and Axially-Symmetric, Circular
Boundary Waveguides

APPLICATION TO GRADED-INDEX SLAB WAVEGUIDES

3.1

3.2

3.3

3.4

Transcendental-Equation Solution for a Slab
Waveguide

Approximate Closed-Form Solution for a Slab
Waveguide

First-Order Corrected Field for a Slab Waveguide

Numerical Data and Results

APPLICATION TO GRADED-INDEX FIBERS

4.1

Transcendental-Equation Solution for Circular
Fibers

Page

10

13

16

23

23

23
25

26

37

Chapter Page

4.2 Approximate Closed-Form Solution for Circular

Fibers
4.3 First-Order Corrected Field fbr Circular Fibers 39
4.4 Numerical Data and Results 39

V. CONCLUSION

REFERENCES

Table

Table

Table

Table

Table

Table

Table

.1.b.

LIST OF TABLES

Comparison of eigenvalue parameter B/k

for high-contrast step-graded-index slab
waveguide obtained by numerical solution
of integral-operator equations, and first-
order, integral-operator-based approximate
perturbation analysis.

Comparison of eigenvalue parameter B/k

for high-contrast step-graded-index slab
waveguide obtained by numerical solution
of integral-Operator equations, and first-
order, integral-operator-based approximate
perturbation analysis.

Comparison of eigenvalue parameters B/k

for high-contrast, a-profile slab waveguides
obatined by numerical solution of integral-
operator equations and first-order, integral-
operator-based perturbation analysis.

Comparison of eigenvalue parameters B/k

for low-contrast, a-profile slab waveguides
obtained by numerical solution of integral-
operator equations and first-order integral-
operator-based perturbation analysis.

Comparison of eigenvalue parameters B/k

for high-contrast, a-profile, graded-inaex
fibers obtained by numerical solution of
integral-operator equations and first-
order, integral-operator-based perturbation
analysis.

Comparison of eigenvalue parameters B/k

for low-contrast, a-profile, graded-index
fibers obtained by numerical solution of
integral-operator equations and first-order,
integral-operator-based perturbation analysis.

Comparison of eigenvalue parameters (B/ko)

for high-contrast, a-profile, graded-index
fibers obtained by numerical solution of
integral-operator equations and first-order,
integral-operator-based perturbation analysis.

Page

27

28

29

40

41

43

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

LIST OF FIGURES

Page

Open-boundary waveguide configuration
consisting of a heterogeneous core immersed
in an infinite homogeneous cladding 5

Description of heterogeneous core by

equivalent sources (p , J ) immersed in the

o e 0 eg 8% e

infinite, homogeneous lad ng medium. 7

Geometrical configuration of slab waveguide
and field components of TE surface-wave
modes. _ 12

Geometrical configuration of circular-fiber
waveguide. 14

Comparison of low-contrast, graded-index

slab waveguide dispersion characteristics

obtained by numerical solution of integral-
operator equations and integral-operator-

based perturbation analysis. 32

Comparison of TE0 eigenmode field distri-

butions on an a-profile, graded-index slab
(low-contrast) obtained by numerical solution

of integral-operator equations with first-order
perturbation field correction. 33

Comparison of TE eigenmode field distributions

on an a-profile, graded-index slab (low contrast)
obtained by numerical solution of integral-
operator equations with first-order perturbation
field correction. 34

Comparison of TE eigenmode field distributions

on an a-profile, graded-index slab (low contrast)
obtained by numerical solution of integral-
operator equations with first—order perturbation
field correction. 35

Comparison of low-contrast, graded-index fiber
dispersion characteristics obtained by numerical
solution of integral-operator equations and the
integral-operator-based perturbation analysis. 44

Figure 4.2.

Figure 4.3.

Figure 4.4.

Figure 4.5.

Comparison of TE01 eigenmode field
distributions on an a-profile, graded-
index fiber (low contrast) obtained by
numerical solution of integral-operator
equations with first-order perturbation
field correction.

Comparison of TE eigenmode field

. . . 0% .
distributions on n a-profile, graded-
index fiber (low contrast) obtained by
numerical solution of integral-operator
equations with first-order perturbation
field correction.

Comparison of TE0 eigenmode field
distributions on An a-profile, graded-
index fiber (low contrast) obtained by
numerical solution of integral-operator
equations with first-order perturbation
field correction.

Comparison of TE eigenmode field
distribution on a-profile, graded-
index fiber (low contrast) obtained by
numerical solution of integral-operator
equations with first-order perturbation
field correction.

Page

46

47

49

CHAPTER I

INTRODUCTION

Of international interest recently are topics embracing the
area of optical communication systems, including fiber and integrated
optics. The implementation of practical systems with optical wave-
guides consisting of low-loss, multimode, graded-index fibers has
motivated intense interest in the research community and a review by
Gloge [l] of lightwave communication studies at Bell Telephone Labora-
tories anticipates that most major trunks on the Bell System will be
converted to optical communication channels by the turn of the century.
Further evidence of this interest is supported by the report by
Schutzman [2] describing the NSF program on optical communication
systems and the survey by Whinnery [3] of integrated-optics research
in the United States. Optical-fiber telecommunications has motivated
the exploration of guided-wave techniques for the construction of
new or improved optical devices. Monomode dielectric waveguides
compatible with monomode optical-fiber systems capable of highbrate
transmission over great distances are primarily utilized by contem-
porary integrated—optics circuits. Among the class of dielectric
waveguides having finite transverse dimensions, however, only the
circular geometry permits an exact analytical solution [4], and then
only for dielectric guides which have spatially-invariant dielectric
properties or those for which the core permittivity is graded in a

simple manner.

Research has been done on the integral-operator analysis of open-
boundary dielectric waveguides, providing new methods for the investi-
gation of contemporary problems involving surface waveguides for optical
and millimeter wavelength radiation [5-7]. This involves the formulation
of an electric-field integral equation (EFIE) for the unknown electric
field in the dielectric-waveguide core; this formulation is ideally
suited to the development of new numerical and approximate (iterative)
analytical solutions. It provides a basis for the study of modal field
and dispersion characteristics through quasi-closed-form solutions,
approximate analytical solutions by way of perturbation analysis, or
direct moment-method numerical solutions. This research investigates
the characteristics of dielectric waveguides by employing two forms of
perturbation analysis approaches.

Typical perturbation analysis [8] uses a zeroth-order eigenmode
which is a known solution to a relatively-simple "unperturbed" problem
as the basis for calculating a first-order "corrected" solution for
the "perturbed" problem. In this study, the "unperturbed" system is a
dielectric waveguide with a constant index of refraction; solutions to
these slab waveguides and circular fibers are well known [9]. "Corrected"
waveguide characteristics and modal fields are then obtained for graded-
index waveguides.

The object of this research is to study the characteristics of
guided surface-wave modes on graded-index waveguides, specifically the
slab waveguide and the circular fiber. A general scalar perturbation
equation is developed which, when applied to the specific cases, provides
a simple computational method for obtaining these characteristics. Two

different perturbation approaches are studied and are found to be

accurate for examining waveguides which contain a small variation of

refractive index within the core.

CHAPTER II

INTEGRAL-OPERATOR-BASED PERTURBATION

ANALYSIS FOR DIELECTRIC WAVEGUIDES

2.1 Equivalent-Current Description for Heterogeneous Dielectric
Waveguides

Heterogeneous dielectric waveguides consist of relatively-high
permittivity, heterogeneous core regions immersed in a uniform, homo-
geneous cladding. An arbitrarily shaped cross-section of a heterogeneous
dielectric waveguide is shown in Figure 2.1. The guide region confines
the radiation by total-internal reflection, causing the wave energy to
be bound to the guiding structure. To formulate the integral-operator
description of such a system, an equivalent current proportional to the
unknown electric field in the heterogeneous core region replaces the
core.

Given'Uun:Maxwell's equations for the EM field (E,fi) maintained

by the impressed current 3e are flO]
v for’) = whom?)
v x m?) = 3%?) + jwefi?) H?) (2.1)

the equivalent current 3eq(;) = jw(e(¥)-ec) E (E) can be obtained by

adding and subtracting the quantity jwec E(¥) to the right side of

’

Ampere's law U1]. Here the complex permittivity e(;) = e’(?) - je’ ($)

2 -+ 2 ‘* -
— n (r) so and 8c = nC so, where n(r) and nC are the refractive

CLADDING ’

e e o /
(Infinite and homogeneous) /
at?) = 6C
e “‘0
CORE
(finite and heterogeneous)
e(‘r’) = e’(?) -j e”('r’)
p" “'0
i
/
-. e. /
(E, )

 

Figure 2.1 Open-boundary waveguide configuration consisting of
a heterogeneous core immersed in an infinite
homogeneous cladding.

indices in the core and cladding. Maxwell's equations are thus expres-

sible as
VxEG) = -jwuoii(¥)
v x E (¥) = 39c?) + Teq(?) + jweCE(?) (2.2)
where 33(E) + jeq(?) = 3t(¥) is the total—effective current maintaining

an EM field (E,R) in a region of effectively homogeneous cladding. 3eq(;)’
which exists only in the core region, can be thought of as representing
polarization currents which augment those currents that would exist in

a homogeneous cladding region in the absence of a core. This is shown

pictorially in Figure 2.2.

2.2 Volume EFIE for Unknown Waveguide Field Excited by Impressed Current
Using the total-effective current 3t = 3e + 3eq’ a solution for the
electric field can be expressed in terms of a linear integral-operator

L as
E?) = M? + 3eq} = EH?) + Fat) (2.3)

where E1 = Lije} is the impressed field maintained by primary, excita-
tory current 3e, and-Es = L{3eq} is the scattered field maintained by

secondary, equivalent-induced current 3e Expressing ES using the

q'
definition of 3eq’ a 3-d linear integral—operator equation (EFIE) for

the unknown E in the core is obtained as

R?) - Liju(e(‘r’)-ec) E(?)} = E165). (2.4)

CLADDING /
(infinite and homogeneous) /

em= EC...forallr x/ >4

 

 

Figure 2.2 Descneiption of heterogeneous core b equivalent sources
loeq' eq) Immersed In the Infinite, omogeneous cladding

medium.

In terms of Lorentz potentials [1d,

' , 2 ++, ,
L{ } = -;%;-IV [V - { } v + kc { }] G(r [r ) av (2.5)

and the volume EFIE can then be expressed as

 

s s JZC
5(r) * kc fvh{V'-[jw(e(¥’) - cc) E(?)] VG(¥|¥’)
+ kcz (jm(e(¥’) - ac) E(¥’) C(FIF’))} dV’ = Ei(?)

...for all r in Vh (2.6)

where Vh is the region of the core where e(;) # Ec’ Zc = Vuo/ec, and

kC = w uoec. C(flf’) is the Green's function for the scalar Helmholtz

equation with R(;,¥’) = If-f’l and is equal to exp (-jkCR)/4nR. With

w2p0(e(?) - cc) = 5k2(?) and noting that [5]

v’ - [5k2(?’) E(¥')] _ V’k2(?’) - E(?')
‘kc ‘ k2

 

- .___75—— n . E(¥’) 6(?’-r ) (2 7)

where T is the boundary contour of the core-cladding interface, the

EFIE can finally be expressed as

v’k2(?’) ' EC?)

k2? C(fl-f’) dV’
h 1")

 

ET?) + V {-fv

2 +, .
+ I My - Ed?) G(f|?’)dS’]

s 2
F kC

-fv 5k2(?’) E(‘f’) cm?) dV’ = E16) (2.8)
h
...for all f in Vh

These three integral terms can be thought of as contributions from
induced equivalent polarization volume charge, surface charge, and volume
current which represent the maintained scattered field. The impressed

. gi -> . . . 3e .
field (r) maintained by exc1tatory current exc1tes the unknown

field E(;) in the waveguide core in this vector EFIE.

2.3 Two-Dimensional EFIE for Natural Surface-Wave Modes

When 38 = E1 = O, the EM fields confined to the waveguide core are
natural surface-wave modes. For an infinite, longitudinally-invariant
waveguide structure the integral over 2’ in equation (2.8) can be
carried out, and a 2-d, vector EFIE is obtained.

A guided travelling surface-wave electric field propagating in the
:fidirection is expressed as E(;) = 3(3) exp(:382), where B is the phase
constant and 3 is the 2-d position vector in the transverse plane. With
a longitudinally-invariant waveguiding structure, n(;) + n(g), and

integrating expression (2.8) over 2’ gives [5]

10

2+ ++ K ( I+ +1)
v.3 (p’) - 8(0’) o Y 9 "

dS’
kzcb”) 2"

 

etc) + (vt + 182) [‘fcs

2 I (YI3 3’1)
+, . < - .
+ o §5——9——-n’ - 3(3’) 0 dt’]
F k 2 2n
C

++,
2.. H Kocho-o I) (2.9)

- f 6k (p’)e(p’) dS’ = 0
CS 2n

 

+ .
...for all p in CS,

_ , +-+’
9:1EEE-dz’ = eIUBZ k0(ylp p D
4nR 2n

used with the definition y = /BZ-kc2.

e+JBZ has been

 

where theidentity f:

If the waveguide core is transversely-graded,the transverse-
component EFIE's uncouple from the longitudinal-component equation
(z-Vt’k2(3’) = O)and equation (2.9) can be written for the m'th guided

eigenmode of the transverse field 3 as

tm

,2+, + +, , + +,
Vt k (p ) - ethD ) K0(lep -p I)

 

 

 

 

e (0) V [f dS’
t t cs ,
"‘ kzc‘p’) 2"
2 +, . K (Y‘Ig-g’l)
_ 6k [p ) , . + +. o m ,
£151‘ k 2 “ etm(p ) 2n d1 1
C
’ ++’
2 +’.+ +’ ko(leo-o I) ’
'fcs 6k (0 )etm(p ) 2” ds - 0 (2.10)

...for all 3 in CS.

This is a convenient form for specializaticntapplications.

2.4 Specialized EFIE for TE Modes of Rectangular Boundary Dielectric
Waveguides

11

For one-dimensional, planar, slab waveguides, the fields are
independent of the transversal coordinate along a tangent to the core-
cladding interface. With fields that are y-invariant, as shown in

Figure 2.3, the integrals over y’ in equation (2.9) can be evaluated

(VIE-K’IMy’ = g. e-le-x I

m

[S]. Noting that f , the l-d, vector

EFIE for a slab guide with the core occupying the region -d :_x :_d

can be obtained as

2 2
+ (n -n ) . . _ _ . . +
eCX) + —-9——55-- [ex(d) (yx-jez) e *(d x) + ex(d) (yx+jsz) e”X d)1
2Yn ”
c
dn2(x')
- §%'[il ex(x') —;%%;T;- (Y sgn (X'-X) x-sz)e-Y|x-x Idx'
k 2
- 7&7 {:1 [n2(x')-nc2] 3(x')e'y{X-x')dx' = 0 (2-11)

...for -d :.x :_d

where nb is the refractive index at the core-cladding boundary.
TE surface-modes for this configuration require 3(x) = yey(x) [5].
These modes are, therefore, obtained as the non-trivial solutions to

the y-component of the above expression for ey(x):

2

k d 2

__o_ le-X'l . _
ey(x) 2y f_d[n dx - O

(x') -nc21 ey(X') e'
(2.12)

...for -d :_x :_d.

12

I’ XA 1'

 

 

 

I

I

: “()0 = n 94* CORE
_! A n th

z/ i V
E'9Ey U CLADDING

Figure 2.3 Geometrical configuration of slab waveguide
and field components of TE surface—wave
modes.

13

Solving this equation when n2(x) is a constant by a Fourier-exponential
transform representation for ey(x) [5] leads to well-known [9] field

soltuion ey(x) = A COS(KX) + B sin (KX) and eigenvalue equation tan(Kd)
= ZYKl/Kz-Yz), where K2 = K2 - 82. The eigenvalue equation reduces to

two independent equations for the even and odd modes, corresponding to

the even and odd parts of the field solution.

2.5 Specialized EFIE for Axially-Symmetric TE Modes of Circular-Boundary
Dielectric Waveguides
Equation (2.9) can be expressed in cylindrical coordinates for
circular-fiber waveguides, as seen in Figure 2-4- In the absence of
axial symmetry, the scalar components are coupled; however, in the sym-

metric case they become uncoupled. The ¢ component equation is

1 2" “2mm: 1 3 ++ ,
e¢(r.¢) + fife ertanb) 7— ;§¢-Koalo-o’l) ado
C
k 2
£71: I02" {-er(r'.¢>') sing-w) + e4, (raw cos (¢-¢')]
Ko(vl3-3'|) r'do' dr' = o (2.13)

BY definition Of a TB eigenmode. ez must vanish and the 2 component of

expression (2.9) requires [5] er(r,¢) = 0. Forcing the r component

14

 

CLADDING
"c
y CORE
4
a r
‘ o
\ n(3)=n(r) x
’ i
" A A
' £1“ = r‘Er + ¢E¢

CLADDING

nc

Figu re 2.4 Geometrical configu ration of circular-fiber waveguide.

15

equation to satisfy this requirement results in e¢(r,¢) = e¢(r).
Therefore, TE modes exist in circular-fiber waveguides only as
axially-symmetric fields, and the ¢ component equation is then

2

k
e¢(r) - 757-4? dr’ [n2(r’) -nc2] r’e¢(r’).l;)2Tr cos(¢-¢’)

xotyIS-S’I) d¢’ = o (2.14)

...for O < r :.a.

Evaluating the angular integral, {3" cos(¢-¢’) Ko(yI3-3’|) do; requires

11' . . .
the change Ko(z) - §'[J JO(JZ) - NO(JZ)]. And so,

 

A?" cos(¢-¢’) Ko(yI3-3’[)d¢’ = a: cos(¢) Ko(y/r2+r’2-2rr’cos(¢))do

 

= ja 4: cos(¢) Jo(jy Vr2+r’2-2rr’ cos(¢)) do

 

- n 4: cos(¢) No(jy /r2+r’2-2rr’ cos(¢)) do.

Integrating the first term by parts gives nJ1(jyr) J1(jyr’); the value

of the second term depends on the relative values of r and r’ and is
«J1 (er) [rtJ’JltJ'Yr’) - Nlmr’m ....for r < r’
"J1 (J'Yr’) [n(ichjvr) - N1(J'Yr))] ...for r’ < r.

Utilizing the transformations K1(z) = j g-[jJ1(jz) - N1(jz)] and

JICIZ) = j11(z), the final 4 component integral equation which describes

16

a TE surface-wave mode is
e (r) - k 2 [K ( r) fr [n2(r’) -n 2] r’e (r’) I (yr’) dr’
¢ 0 1 Y o c ¢ 1

+ 11(YT) I: [n2(r’) - ncz] r’e¢(r’) K1(yr’) dr’] = o
(2.15)

...for 0 < r < a.

Again, as in the rectangular boundary case, solving this equation
for a constant refractive index by a Hankel transform leads to the
correct 9 field = AJ ' i '

[ ] e¢(r) 1(Kr) and eigenvalue equation J1(Ka)/(KaJo(Ka)
= - Kl(ya)/(ya)Ko(ya), where)? = kz-BZ.

2.6 Development of Perturbation Equation for TB Modes of Rectangular
and Axially-Symmetric, Circular Boundary Waveguides
In order to develop a generalized, scalar perturbation solution
based upon the electric-field integral equation for rectangular and
axially-symmetric, circular-boundary dielectric waveguides, the follow-
ing notation is adopted (here the subscript 'm' refers to the m'th

natural mode):

==> wmcu)

) ==> Km(u|u’)

 

l , + +, ,
-7; 4," cos(¢-¢ ) KOCYmIo p I) d¢ ‘

l7

) ==> I: du’

 

a , ,
fo r dr J

This allows the EFIE's for the rectangular and axially—symmetric,

circular TE surface-wave modes, (equations (2.12) and (2.13))

-Y Ix-X’l
erCX) -§$;-{i1 6k2(x’) eym(x’) e m _ dx’ =

1 a . , 2 , , 2a . +_+, ,
e¢m(r) - EF'fl) dr r 6k (r ) e¢m(r ) 4) cos(o-¢ ) Ko(yml p |)d¢
to be written as the general scalar equation
v (u) - rb 6k2(u’) v (u’) K (u|u’) du’ = o (2 15)
m a m m ' ' °

This equation can be even more conveniently expressed if the operator

acting on the m'th electric field mode is defined as:
L { } = fb 5k2(u’) { } K (u|u’) du’
m a m '

Expression (2.15) is then written as

Wm(u) - Lm{wm(u)} = 0. (2.16)

18

Suppose that a solution with Wm(u) = Vm(0)(u) and Ym = Ym(o) is

known when 5k2(u) = Ak2(u). Vm(0) is then defined as the solution of

 

Vm(0)(u) - Lm(°) {wm(°)(u)} = o (2.17)
where
Lm(0){ } = 4? du’ Ak2(u’) { } Km(0)(ulu’)
and
(0)
r 1 e-Ym |x-x’|
ZY (0)
K (0) m

m (ulu’) => i + +
71;,— Jzoflcos (¢-¢’) K0(Ym'(0)Io-p’l) dt’

 

L

Vm(0)(u) and Ym(0) are defined as the electric field and wavenumber

parameter, respectively, of the m'th unperturbed mode. A perturbation

2
p'(u).

Similarly, the first-order perturbation equation can be written

solution is now sought for the case of 6k2(u’) = Ak2(u) + 6k
as
vm(1)(u) - Lm(1) {vm(1)(u)} = o (2.18)
where
Lm(l) { } = (f duf5k2(u’) { } Km(1) (ulu’)

and

19

-Ym(1)|X-X’|

Km(1)(ulu’) = >4

 

1 2n , .(1) +.+’ ,
'-- Q) COS(¢-¢ ) K0(Ym Io-p I) d¢ -

As in conventional analysis in perturbation theory [8], a term which
involves the unperturbed operator acting on the first-order perturbed
field, Lm(o){wm(1)(u)}, is added and subtracted to equation (2.17), and

the following expression is obtained
(1) _ (0) (1) _ (1) (1) -
vm (u) LIn {Tm (u)} Qm {Tm (u)} - 0 (2.19)
where
(1) _ (1) _ (0)
Qm{}-Lm {11m {1
The first-order corrected field can be expressed as a small correction
term added to the zeroth-order field: Wm(1)(u) = Wm(0)(u) + 6Wm(1)(u).
When this is substituted into the above expression, and definition
(2.17) is recognized, equation (2.19) becomes
(1) _ (0) (1) _ (1) (0)
sum (u) Lm {avm (u)} Qm {vm (u)}

- Qm(1){6vm(1)(u)} = o. (2.20)

However, only first-order correction terms are being considered here,

20

and since Qm(1){6Vm(1)(u)} is a second-order correction, equation (2.20)

can be simply written as
6Vm(1)(u) - Lm(°){avm(1)(u)} - Qm(1){vm(°)(u)} = o. (2.21)

Operating on the above equation with

r
rdddmkzcxi em”) (x)

fb du Ak2(u)V (0)(u) = >4
3 m 2 (0)
foadrr Ak (r)e¢m (r)

 

L

gives
4? du Ak2(u) vm(°)(u) svm(1)(u)
- 4f du Ak2(u)vm(°)(u) 4f du’ Ak2(u’) 6Vm(l)(u’) Km(0)(ulu’)
- Q? du Ak2(u)wm(°)(u) chl) {vm(°)(u)} = o. (2.22)

Consider now the second term of equation (2.22); it can also be written

as
4? du Ak2(u) evm(1)(u) 4f du’ Ak2(u’) vm(°)(u’) Km(0)(u’|u)

It is important to recognize that the kernel function Km(u|u’) is

symmetric and equal to Km(u’lu). This is obvious for the slab waveguide,
. 1 WmIX-x’l
where Km(x|x ) = §;—-e . For the circular waveguide, the kernel
m

21

1

. . . . 2n +
function, Km(r|r ), is 2?' 4) p

cos(¢-¢’) Ko(ym| -3’|) do’ and is

readily seen to be symmetric after noting that

 

'343’I = /r2 + r’2 - 2rr’ cos(¢-¢’)

And so equation (2.22) can be written in the following form:

b

I: du Ak2(u) 5vm(1)(u) [vm(°)(u) - [a

du’ Ak2(u’) wm(°)(u’)Km(0) (ulu’)]
b
- fa du Ak2(u) vm(°)(u) Qm(1) {vm(0)(u)} = 0.

(2.23)

Again, realizing that definition (2.17) is embedded in the first term

of the above expression, equation (2.23) becomes

4? du Akzcu) wm‘°)(u) (Lm‘l) {wm(°)(u11 - Lm(°){wm(°)(u)11 = o.

(2.24)

Equation (2.24) is now in a form that allows the first-order cor-
rected wavenumber parameter, ym(1), to be solved for, once the unperturbed
fields are known. This is more clearly apparent when the above expres-

sion is written as
{f du Ak2(u) (vm(°)(un3

= if du Ak2(u) vm(°)(u) r? du’6k2(u’) vm(°)(u') Km(1)(u|u’) (2.25)

22

which is a transcendental equation for Ym(1) embedded in Km(1)‘ And
finally, in the special case where Ak2(u) is considered to be constant,

the following transcendental equation is obtained

If c.1u (11mm) (11))2

= 4? du vm(°)(u) 4? du’ 6k2(u’) wm(°)(u’) Km(1)(u|u’). (2.26)

Either of these results can be used to solve for the first-order cor-

rected wavenumber parameter by knowing only the unperturbed fields.

CHAPTER III
APPLICATION TO GRADED-INDEX SLAB WAVEGUIDES

3.1 Transcendental-Equation Solution for a Slab Waveguide

Two different perturbation solutions have been employed to
determine the first-order, corrected wavenumber parameter. One
method is the transcendental-equation solution. Expression (2.26) is
solved for ycl) embedded in the kernel function (model-index m is sup-
pressed for brevity). When written for the slab waveguide, that

equation becomes

{it [ey(0)(x)]2 dx
___i_ ,d dx 8. (mm ,d 61‘2“.) e (0)0” e-Y(1)lx-X’ldx,
2Y(1) 'd Y 'd y '

(3.1)

ey(o)(x) = A cos (KX) + B sin (xx) is known (Section 2.4 and [9]).
Equation (3.1) can then be numerically integrated and, when forced to
be satisfied, provides a transcendental equation for the first-order

(1).

eigenvalue y

3.2 Approximate Closed-Form Solution for a Slab Waveguide
Another method of finding the unknown first-order eigenvalue is

(0) (1) (0)

to express y(l) as y + Av . y is the exact eigenvalue when

ey(x) becomes the exact eigenfield solution, ey(0)(x), for the

23

24

"unperturbed", uniform, step index slab where the refractive index is

a constant within the core, n2(x) = n02. For a small correction compared

Y(1)
-l—- can be approximated as -%fj(l-A :(0))’ and

Y
(1) M(1)|

1)
for a small Av , e x I = l-AY(1)Ix-x I. Using these approxi-

mations, equation (3.1) becomes

{i1 [ey(0)(x)]2 dx

Mm
-—(——20) (1- "7—7) dx ey(0)(x) {dd ahzot’) ey(o)(x’)
(0) ,
-7 lx-x I (l-Ay(1)Ix-x’[)dx’. (3.2)

This can be expanded out, with 6k2(x) = Ak2(x) + 6k20(x) where Ak2(x)

_ 2 _ 2 2 2 2 _ 2 2 _ 2 .
— Ak - ko (no -nc ) and okp (x) - ko (n (x) no ), and the terms which

(1)/Y

(0)) is assumed small

(0) (x)

involve (AY(1)/y(OD2 are dropped, since (A7
at the outset. Also, it is noted that (see equation (2.17)) ey

is defined as

e (0)00 = 93—2.— d -y(0)|X-X’I dx’

(0) ,
f e (x ) e
2Y(0) 'd Y

and that using this definition, the term on the left of equation (3.2)
is repeated by the leading terms on the right hand side. Collecting

(1)

the remaining terms and solving for Av gives

25

(0) ,
Y -d x y -d Y .

N

+ Ak Add dx ey(0)(x) {dd ey(0)(x’) e‘7

(0) ,
+ y(0) iii dx ey(0)(x) {it 6k02(x’) ey(0)(x’)Ix-x’| e'Y Ix-x I dx’

(0) ,
+ {$1 dx ey(0)(x) {i1 dkpz(x’) ey(0)(x’) e-Y Ix-x I dx’)
(3.3)

This expression can be computed by numerical integration in terms of

(0)

the zeroth-order eigenmode quantities y and ey(0)(x), and an approxi-

mate closed-form solution obtained.

3.3 First-Order Corrected Field for a Slab Waveguide

Using equation (2.12) a first-order, integral-operator-based
perturbation field correction can be obtained. Letting the field take
its "unperturbed" uniform-slab value, but grading the index gives the

expression

1 I‘d ym) Ix-x’l
(0) -d

 

e (l)(x) = 6k2(x’) e (O)(x’) e- dx’

(3.4)

26

which results in a first-order corrected field. A further correction

(1):

can also be obtained by using the first-order eigenvalue 7

1
(1)

 

(1) _»
v vax | dx’

6k2(x’) ey(0)(x’) e‘ (3.5)

ey(1)(x) = -d

27

3.4 Numerical Data and Results

The first try at getting a corrected wavenumber parameter for the
slab waveguide was finding Ay(1) by enforcing satisfaction of the integral
equation of a step-graded-index waveguide using the O'th-order eigenfield.
Equation (3.3) was point matched at different points 'x'; that is, the
second integration with the "unperturbed" eigenfield weighting was not
done. These values were then averaged; Table 3.1 shows them compared
with the direct numerical solution and the double-integrated closed-form
solution. After seeing that the averaged point-matched value was very
reasonably close to the exact numerical solution, and that the double-
integrated weighted value was in most cases even closer, the approxi-
mations made for the closed-form solution seemed appropriately correct.

Tables 3.2 and 3.3 show results for high-contrast and low-contrast
a—profile, slab waveguides, where n(x) = no [l-AIx/dla]k. Perturbation
solutions are compared with accurate (potentially-exact) direct numerical
solutions obtained by the pulse-function expansion of the discretized
(point matched) corresponding integral equation (2.12). The
transcendental-equation perturbation solutions follow very closely
the accurate numerical values, whereas the approximate closed-form
solution tends to become more inaccurate for the higher modes. This

-AY(1) Ix-

can be predicted by examining the approximation e x I: l-Ay(1)|x-x’l.

Table 3.1.b. Comparison of eigenvalue parameterfi/kO for
high-contrast step-graded-index slab waveguide
obtained by numerical solution of integral-
operator equations. and first-order. integral-

operator-based approximate perturbation analysis.

 

 

 

 

 

 

 

n
c
”a x = d
1.05 no 1” = I 0
n
O --
”c x — d
d/xo = 0.3 {
5(0),,(0 5(1),,(0 p(l)/k0 3(1),,(0
MODE NUM NUM PPM PTE
TEO 1.492 1.547 1.539 1.541
TEl 1.164 1.182 1.186 1.181

 

 

 

 

 

 

NUM: numerical solution of integral-Operator equations

PPM: perturbation. averaged point-matched solution
PTE: perturbation. transcendental-equation solution

HIGH-CONTRAST SLAB

28

Tab1e3.2 Comparison of eigenvalue parameters fl/ko for
high-contrast.ia-profile slab waveguides
obtained by numerical solution of integral-Operator
equations and first-order. integral-Operator-

based perturbation analysis.

 

 

 

fiVkO fiVkO BYKO fiVkO
d/AO MODE 0': a: a: “=2
NUM NUM PTE PCF
TEO 1.587 1.537 1.493 1.476
TEl 1.547 1.404 1.339 1.228
1.1
152 1.479 1.264 1.248 1.114
153 1.380 1.117 1.129 CUTOFF
0.15 TEO 1.349 1.218 1.217 1.159

 

 

 

 

 

 

 

 

NUM: numerical solution of integral-operator equations
PTE: perburbation. transcendental-equation solution
PCF: perturbation. approximate closed-form solution

”0 = 1.6
HIGH-CONTRAST SLAB

n 1.0

C
29

Table2343 Comparison of eigenvalue parameters fiVkO for
low-contrast. a-profile slab waveguides
obtained by numerical solution of integral-operator

equations and first-order integral-operator-
based perturbation analysis.

 

 

 

 

 

 

 

 

 

 

13/ k0 13/ k0 fl/ko 5/ k0
d/AO MODE a: a: 0:2 or:
NUM NUM PTE PCF
TEO 1.00970 1.00875 1.00807 1.00774
9 0 TEl 1.00879 1.00625 1.00531 1.00338
TEZ 1.00730 1.00377 1.00356 1.00118
1E3 1.00525 1.00139 1.00149 CUTOFF
1.5 TED 1.00581 1.00379 1.00375 1.00289
NUM: numerical solution of integral-Operator equations
PTE: perturbation. transcendental-equation solution
PCF: perturbation. approximate closed-form solution
110 - 1001
Low-CONTRAST SLAB
nC = 1.00

30

 

31

For example, the lowest mode for the low-contrast waveguide has

_ (1)
Ay(1)d = -.3967, and e AV d = 1.4869 is fairly close to 1 - Ay(1)d
= 1.3967; however, the highest mode has a value for Ay(1)d = 2.8158

AY(1)d (1)
= 16.706 is not at all fairly approximated by 1 - Av d

and e-
= 3.8158. This discrepancy is even more pronounced for the high modes
of the high-contrast waveguide. It should be noted, though, that even
for the low-contrast waveguide an a = 2 profile is very strongly

graded and does not represent a "small" perturbation, and better closed-
form solutions would be expected with higher-a profiles.

Figure 3.1 shows the dispersion characteristics for a low-contrast
slab. The transcendental-equation solutions are almost equal to the
accurate numerical values, even for the thicker waveguides and higher
modes. The closed-form perturbation solution becomes slightly less
accurate for the thicker waveguides (again, the approximation

-Ay(1) (1) . .
e = l-Ay used in the closed-form solution development would
be less accurate for larger d), and are quite inaccurate for the higher
mode.

Figures 3.2, 3.3, and 3.4 are comparisons of the eigenmode field
distributions obtained by the ‘nwmerical solution of the integral-
operator equations and the first-order perturbation field correction
using equations (3.4) and (3.5). Both Figures 3.2 and 3.3 are field
distributions of the TE0 modes and use equation (3.5) for the first-
order perturbation field correction; that is, the first-order eigenvalue,
y(1), is used with the "unperturbed" eigenfield, ey(0)(x), to obtain a
first-order correction. For the thinner waveguide, d/A = 1.5, the

perturbation of the field distribution is slight, and the first-order

correction reproduces the numerical solution distribution. However, the

normalized propagation phase constant B/ko

D
II

LOW-
SLAB

1.010

1.009
1.008
1.007
1.006
1.005
1.004
1.003
1.002

1.001
1.000

32

 

" 1-01 a=0 numerical solution of

1.00 —-—— «=2 integral-operator equations

0 O O perturbation. transcendental-
CONTRAST equation solution (¢=2)

perturbation. approximate
A A A closed-form solution («=2)

 

 

 

 

 

      

 

 

      

0.5 "1.0 1.5 2.0 2.5 3. 3.5
normalized slab-waveguide thickness d/AO

0.0

Figure 3.1 Comparison of low-contrast. graded-index slab

waveguide dispersion characteristics obtained
by numerical solution of integral-operator
equations and integral-operator-based
perturbation analysis.

1.01
1.00

no

nc

LOW-CONTRAST
SLAB

33

 

(cI= 0)

-"--(a=m

O OO(°'=2)

 

 

 

numerical solution of
integral-operator
equations

first—order. integral-
operator-based pertur—

 

   
 
 

      

 

 

bation . field
correction
d/xo = 1.5
1'0 I I *1 T r I
47k = 1 00581
.. 0- '
g 0.8 x -1
0E
\ 006” ‘1
E3
09>. OILIF- ) ‘4
a» 10
$3 0.2.. _
g; 0.0.. 1
'0
33 -0.2_. _
a:
“D
E -an_ .1
E -0'6' TEO mode '1
C
‘0-3- n(x) = n0 [l-AIx/dfll/2 -
-1.0 1 1 1 1 1 L 1 L 1
-100 -008 -006 -034 -002 0.0 002 00“ 0a6 008 1.0
normalized slab-waveguide variable x/d
Figure 3.2 Comparison of TEO eigenmode field distributions

on an a-profile. graded-index

slab (low-contrast)

obtained by numerical solution of integral-Operator
equations with first-order perturbation field

correction.

n = 1.01
n = 1.00

LOW-CONTRAST
SLAB

34

 

(0 = 0) Numerical solution of
... _ ... (a = 2) integral-operator
equations

0 O O (a= 2) first-order. integral-
operator-based pertur-
bation field correction

 

 

 

    
 

 

 

1'0 r T I I ‘r'
19/ = 1.00970
0.8.. k0 _
e
E
‘D 0.6
\
3 0.4
03>.
(1)
”g. 0'2 / . \
*2 10
a 0.0., / \ ...,
E
D
2 020 ’ 01.00807.
a.) " - "‘ "
8 -an_ "
£3
g_0.6_ TEO mOde _
:3 1/2
-008— — —
_ n(x) - n0[ 1 Zl|xld|] ‘
-1.0 1 1 1 J, 1, 1 1 1 1
'100 -008 -006 -002 000 002 004 006 006 008 100
normalized slab-waveguide variable x/d
Figure 3.3 Comparison of TEO eigenmode field distributions on

ancx-profile.'graded-index slab (10w contrast)
obtained by numerical solution of integral-operator
equations with first-order perturbation field
correction.

 

3S

 

no = 1'01 (6= 0) numerical solution of
n = 1.00 .— ... .— (a= 2) integral-operator
C equations
LON-CONTRAST O O O O’th 0rd. egnval. 1'st order
SLAB A a a l’st ord. egnval. perturbation
field correction
_ (a = 2)
d/Ao - 900
1.0 r ' " I l r T I
e» .a

0.8 ._ 15 4‘ _
é '/ \
(D
\ 0-5 " ./ ‘ 0 1.00870 '*
32
;> 0.4 '- J \A

.a

E; 0h2 r-.// .A A. If
'3 0.0 — ..
g 8 1.00531 0 ‘
I, w
E -0'2 — A A /‘
C \5/k0 = 1.00879 /
IE -0.4 " .A qp ‘
E -05 _. \ 1.00625/. ..
g 0 8 Tel mode \ ‘ /

- . — . -

n(x) = nO [l-AIx/dl] 1/2 \ .
_ 1 1 L l l 1‘ . fx. 1

 

 

00
'100 -008 -006 '00“ '002 000 002 004 006 008

 

 

1.0

normalized slab-waveguide variable x/d

Figure 3.4

Comparison of TEl eigenmode field distributions on

an a-profile. graded-index slab (low contrast)
obtained by numerical solution of integral-Operator
equations with first-order perturbation field

COFFGCUOD.

36

thicker waveguide, d/A = 9.0, has a more pronounced a = 2 deviation

and the first-order correction does not follow the numerical solution
very well, especially at the boundary. For the TE1 mode_of the thicker
waveguide, both equations (3.4) and (3.5) are graphed for the first-
order corrected field distribution. Notice that the first-order
eigenvalue solution follows the direct numerical solution better near
the center of the slab, whereas the zeorth—order solution follows more
closely near the boundaries. Since ey(x) is weighted by 6n2(x) k02 in
the integral equation (2.12), the errors in ey(x) near the core/cladding
boundary are relatively unimportant because 6n2(x) is very small there;
the "better" solution is therefore that which is most accurate in
regions where the index contrast is large - the first-order eigenvalue
solution. Again, this is a thick waveguide and the perturbation is
large; the perturbation theoretical development assumes small pertur-
bations, and it is not surprising that the first-order perturbation
field distributions are inaccurate. The excellent results obtained

for the thinner waveguide with a small field deviation validates the

perturbation approach for small perturbations.

CHAPTER IV
APPLICATION TO GRADED-INDEX FIBERS

4.1 Transcendental-Equation Solution for Circular Fibers

Equation (2.26) has the first-order corrected wavenumber parameter
y(1) embedded in the kernel function. This expression can be written
for axially-symmetric, graded-index circular waveguides as

1.3

o [e¢(0)(r)]2 rdr

= A? rdre¢(0)(r) K1(y(l)r) a: 5k2(r’) e¢(0)(r’) 11(y(1) r’) r’dr’

+ (f rdr e (O)(r) 11(y(1)r) a? 6k2(r’) e¢(0)(r’) K1(Y(1)r’) r’dr’.

¢
(4.1)

As found in Section 2.5, the well-known [9] zeroth-order field for the
"unperturbed" step-index fiber is AJ1(Kr). Forcing the above equality,
as in the case of the slab waveguide, provides a transcendental equation

(1).

which determines the value of y

4.2 Approximate Closed-Form Solution for Circular Fibers

7(1) is obtained from its definition

To get a closed-form solution,
as yCO) + Ay(1). Kl(y(0)r + Ay(1)r) is then approximated by the leading
terms of it's Taylor's series as K1(y(0)r) + Ay(1)r K1’(y(0)r) and
Il(y(0)r + Ay(1)r) = 11(Y(0)r) + Ay(1)r Ii(y(0)r). Substituting these

approximations into equation (4.1), and invoking (see definition (2.17))

37

38

(0)
¢

the defining integral equation for e

e¢(0)(r)
= Ak2 [K1(y(0)r) (I e¢(0)(r’) 11(y(0)r ’) r dr’
+ 11(Y(O)T) g? e¢(0)(r’) K1(Y(O)r ’) r dr 1

the left side of equation (4.1) is duplicated by leading terms on its
right, and all terms involving (Ay(1))2 are dropped (assumed negligibly
small). The remaining terms are then arranged to solve for the first-

order correction to the wavenumber parameter, Ay(1), in a closed-form

M0)
= (-(f rdr e¢(0)(r) K1(Y(0)r) 4f akzpcr’) e¢(°)(r’1 11(y(°)r ’1 r dr

- a? rdr e¢(0)(r) I1(Y(O)r) {: 6k20(r’) e¢(0)(r’) Kl(y(o)r ’) r’dr’ )/

(4? rdr e¢(0)(r) K1(y(0)r) a: 6k2(r’) e¢(0)(r’) Il’(y(0)r’) r’2 dr’

+ foa r2 dr e¢ (0)(r) K1’(y (O) r)fH6k (r’ ) e¢ (0)(r’ ) I 1(y(0)r’) r ’dr’

+ 4? rdr e¢(0)(r) 11(y(0)r) A? 6k2(r’) e¢(0)(r’) Kl’ (y (0) r’) r’ 2dr’

+1: r2 dr e ¢(O)(r) 11(y (O)r) (f 6k2(r’) e¢(0)(r’) K1(y(0)r’) r dr )

where Skpz (r) and dkz (r) are as defined in Section 3.2. 14-2)

39

4.3 First-Order Corrected Field for Circular Fibers
Again, as for the slab waveguide, a first-order, integral-operator-

based perturbation field correction is obtained for the circular fiber

(0)
¢

eigenvalue, 7(0), in equation (2.15) for the perturbed fiber. This

by using the zeroth-order field, e , together with the zeroth-order

results in
e¢ (l) (1.)
K1 (y(o)r) 4f 6k2(r’) e¢(0)(r’) 11(y(0)r’)'r’dr’
+ 11(y(0)r) a? 6k2(r’) e¢(0)(r’) K1(Y(O)r’) r’dr’. (4.3)

(1)

If the first-order eigenvalue 7 is used, the expression becomes:

e¢(1)(r)

= K1(y(l)r) 4f 6k2(r’) e¢(0)(r’) Il(y(1)r’) r’dr’

+ 11(y(1)r) a? 6k2(r’) e¢(0)(r’) K1(y(l)r’) r’dr’. (4.4)

4.4 Numerical Data and Results

Many of the results acquired from the slab waveguide are replicated
for the circular fiber. Tables 4.1 and 4.2 show comparisons of the
eigenvalue parameter 8 for graded-index fibers obtained by the numerical

solution of the integral-operator equations and by the first-order,

40

Table 4.1 Comparison of eigenvalue parameters p/ko for high-
contrast. a-profile, graded-index fibers obtained by
numerical solution of integral-Operator equations
and first-order. integral-Operator-based perturbation

 

 

 

 

 

analysis.
B/ko p/ko B/ko B/ko
a/x0 MODE. a=0 a=2 a=2 a=2
NUM NUM PTE PCF
TE01 1.570 1.459 1.369 1.289
T602 1.497 1.304 1.289 1.218
1.85
. TE03 1.377 1.129 1.150 1.102
TED“ 1.198 CUTOFF CUTOFF CUTOFF
TED] 1.461 1.254 1.213 1.123
0.8 '
T502 1.101 CUTOFF CUTOFF CUTOFF
0.69 TEOl 1.419 1.193 1.166 1.076

 

 

 

 

 

 

 

NUM: numerical solution of integral-Operator equations
PTE: perturbation. transcendental-equation solution
PCF: perturbation. approximate closed-form solution

no =1.6
HIGH-CONTRAST FIBER

”c = 1.0

41

Table 4.2 Comparison of eigenvalue parameters p/ko for low-
contrast. a-profile. graded-index fibers obtained by
numerical solution of integral-operator equations
and first-order. integral-Operator-based perturbation

 

 

 

 

 

 

 

 

 

 

 

 

analysis.
B/ko alko B/ko filko
a/xo MODE: a=0 a=2 a=2 a=2
NUM NUM PTE PCF
T501 1.00929 1.00701 1.00548 1.00404
TE02 1.00763 1.00402 1.00388 1.00276
15.0
TE03 1.00506 1.00109 1.00135 1.00088
TED“ 1.00169 CUTOFF CUTOFF 1.00004
T501 1.00821 1.00503 1.00411 1.00283
9.0 '
TE02 1.00413 1.00038 CUTOFF 1.00020
5.0 T501 1.00525 1.00137 1.00117 1.00022
NUM: numerical solution of integral-operator equations

PTE: perturbation. transcendental-equation solution
PCF: perturbation. approximate closed-form solution
n0 = 1.01‘
l Low-CONTRAST FIBER
nC = 1.0

 

42

integral-operator-based perturbation solutions for high- and low-contrast
fibers, where n(r) = nOII-A(r/a)a]k. The transcendental—equation pertur-
bation solution follows the numerical values very well for the low-contrast
fiber and well for the high-contrast one. Again, as for the slab wave-
guide, the approximate closed-form perturbation solution values deviate
from the numerical ones, especially for the high-contrast fiber. a = 2

is a large perturbation profile and the approximations made for the

(1)

closed-form solution fail to be valid approximations when Ay gets large-

For example, when a/A = 9.0 the TB mode has AyCl) = 1.582 and the

(1))2

01
= 2.5 is not negligible as assumed in the closed-form

term (Ay
development, nor is only the leading term of the Taylor's series expan-
sion sufficient. Table 4.33hows that for a small perturbation profile,
a = S, that even for the high-contrast fiber the closed-form perturbation
solutions are in close agreement with the numerical-solution values, and
so the approximations made for small perturbations are justified.

Figure 4.1 shows a comparison of the dispersion characteristics
obtained by the numerical solution of the integral-operator equations
and the integral-operator-based perturbation solutions for the low-
contrast, graded—index fiber. As for the slab waveguide, the transcendental-
equation solutions are in close agreement with the numerical solutions,
although as the radius increases so does the discrepancy. The closed-
form solutions are again quite inaccurate. This is due to the large
perturbation and the inaccuracies thus inherent. Also, hybrid modes
are the lowest modes for a circular fiber, and so the higher—order TE

modes predictably are more erroneous, since the results became more

inaccurate for the higherg modes in the slab waveguide case.

43

Table 4J3 Comparison of eigenvalue parameters (B/ko) for high-
contrast. a—profile. graded-index fibers obtained by
numerical solution of integral-Operator equations and
first—order. integral-operator-based perturbation

 

 

 

analysis.
0/k0 B/ko B/ko 87k0
o/xo MODE a= a=5 a=5 a=5
NUM NUM PTE PCF
1501 1.570 1.536 1.498 1.484
1502 1.497 1.418 1.392 1.374
1.85
1503 1.377 1.257 1.257 1.235
1504 1.198 1.053 1.058 1.066

 

 

 

 

 

 

 

NUM: numerical solution of integral-operator equations
PTE: perturbation. transcendental-equation solution
PCF: perturbation. approximate closed-form solution

['10 =l|6
HIGH-CONTRAST FIBER

44

 

 

“0 = 1.01 cr=0 numerical solution of
"c = 1.00 ...—.... a=2 integral-Operator equations
0 O O perturbation. transcendental-
Low-CONTRAST equation solution «1=2)
FIBER £5 A: A: perturbation. approximate
closed-form solution «1=2)
1.010 I l I I I I I
1
c: _ - ..
g 1'009 n(r) = n0[1-A(r/o)°‘]2
:5 1.008—
53
m /
c: P / _
8 1.007 TEOI //
cu /
8 1006- ~
/’ -
8 1.005— / . 0
g; / . /
o 1.004- TE / A 1
53‘ 0y ' TEO V
Q 1003- /’ TE y A -
8 02 y A
i“— /o A ..
.—. 1.002 - /O’
c: ./ A:
E
E? 1 001 “‘ A: ‘
' 1' 2. n / 6.17 /A
1.000 I ‘— ' ' ‘

 

 

 

 

l
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0
normalized waveguide radius a/AO

Figure 4;1 Comparison of low-contrast. graded-index fiber
dispersion characteristics obtained by numerical
solution of integral-Operator equations and
the integral-Operator-based perturbation analysis.

45

Figures 4.2, 4.3 and 4.4 indicate TE eigenmode field distribution

01
comparisons obtained by the numerical solution of the integral-operator
equations and the first-order perturbation field correction procured

from equations (4.3) and (4.4). For the T501 mode, it can be seen that
as the radius increases, the perturbation field solution based on the
first-order eigenvalue becomes more removed from the numerical solution
due to the greater perturbation and the larger deviation of the perturbed
field from the unperturbed field. For the TEO2 eigenmode, both the
zeroth- and first-order eigenvalue first-order field correction distri-
butions are plotted in Figure 4.5. Similar to the slab waveguide case,
the first-order eigenvalue correction more closely approximates the
numerical solution near the center of the guide, but deviates more near
the core/cladding boundary where the zeroth-order eigenvalue field
correction is more accurate. Again, since in integral equation (2.15)
e¢(r) is weighted by 8n2(r), and the difference between core and cladding

refractive indices is less pronounced at the boundary, the first-order

eigenvalue soltuion produces a "better" first-order field correction.

46

”0 = 1.011 ---- (a=0) numerical solution

, ._______ (a=2) of integral-Operator

 

 

 

 

 

 

equations
nC = 1.00
‘ o o o (a=2)tgirgt-05der.tingegggrlf
opera r- ose per ur a
%?§E%ONTRAST field correction
O/KO = 5.0
1.0 l 1 1 I -" 1 1
/ "

. ax ..
r 8
3 0.8 .— J \ .4
Z: 1.
‘1} 1.00137/' \\
a) .- B/ko = 1.00525 ,\ -
8 7 \
:J
E. 0.6 ” / ' \ "
8 f . \
f; t 1.00117 \
a / ‘
:2 0.4 - ,1 -
t, in
‘” /
2 __ TE mode -
'“ 01
E 4 a 1 ..
8 0'2 _/ n(r) = n0[l—A(r/a) ]2

0.0 1 1 1 1 1 1 1 1 1
0.0 0.2 0.4 0.6 0.8 1.0

normalized radial variable r/a

Figure 4.2 Comparison of T501 eigenmode field distributions on
an a-profile. graded-index fiber (10w contrast)
obtained by numerical solution of integral-Operator
equations with first-order perturbation field
correction.

47

 

 

 

 

 

 

n0 = 1.01 (a=0) numerical solution
(a 2) of iptegrol-ODerator
- """"" = equo ons
nc - 1000
' ' ' ““21 1.1215808“ 1089101. ..
- - use per ur a on
kngEONTRAST field correction
U/KO = 9.0
1.0 r* 1 ‘ T l 1
/ O \ O
r- . -
1.00503/ ’ \
g 0.8" \ o 2
a) / O \
i:
t: ' / \\ o» ‘
f}
a) 1. ‘\
§ 0,5- / fllko =1.00821 \ . -
i /
'53 ~ /. \ 1..00411 4
c / \
33 0.4 c \\ .’
t __ I. \ _
13 lo
f§ I T501 mode \\
z; 0 2 —u/ 1 ‘\\ a» -
E5 _
‘g 1 n(r) = n0[1-A(r/0)a]2 \
0.0 1 1 J 1 L 1 1 1 1
0.0 0.2 0.4 0.6 0.8 1.0
normalized radial variable r/a
Figure 4.3 Comparison of T501 eigenmode field distributions on

an a-proflle. graded-index fiber (10w contrast)
obtained by numerical solution of integral-Operator
equations with first-order perturbation field
correction.

48

 

 

 

 

 

 

"0 = 1.01 (0=0) numerical solution
_______ (a=2) of integral-operator
nC = 1.00 equations
_ o o o (a=2) first-order. integral-
EQgEfiONTRAST operator-based perturbation
field correction
O/KO = 1510
l.(‘ I I . 1 I r
_ / '\ .
x / - \ .
° /
0E 0.#> \ "
E: / 10 o»
e _ \ .
.39 I \
o)
:23 0.61- / \ 9
:3 . \ 1.00548
:2 __ I .
E I . B/ko = 1.00929 \ ‘*
E 0 \ . ..
c “f I \
8 .- I \ . ..
g I \
g 0.2L] TEol mode \\ .
o 1.
C I n(r) = no[1-A(r/o)<>L]2 \ . _
\\ in
'\~
0.31 1 1 1 1 1 1 1 T3}
0.0 0.2 0.4 0.6 0.8 1.0

normalized radial variable r/a

Figure 4.4 Comparison of TEol eigenmode field distribution on
anoeprofile. graded-index fiber (10w contrast)
obtained by numerical solution of integral-operator
equations with first-order perturbation field
correction.

”0 =1.01

nc

LOW-CONTRAST

1.00

 

49

(a=0) numerical solution of
integral-operator
equations

 

-—----(a=2)

o o o 0'th 0rd. egnval. 1'st-order pertur-
“ bation field
‘ ‘ 1’st ord. egnval.

 

 

 

 

 

FIBER correCtion 50:2)
G/KO = 15.0
1.0‘ ‘.V' 1 1 l l 1 71*
0.8 - / \ B/ko = 1.00763 -
X

08 0.6— 1 \ _ -
> / 1.00402 *
:9 0.4- \ -
.. / ‘\ 8
g 0.2 ' \ 100038 —
=3 .s ‘*.s ‘k
:3 1L4 A. .A
CI; 0.0 : IP I : . IL I ml
5 \ ‘ 1.00763 )/“
v \ ' o ’ 4
z; -0.2’ A ' 0 /
a: ‘1 ‘ //
8 -0.“ " TE02 mOde \ ‘ "'
N
g \ / _
E -0.5 " \__/
8 Gt l

-0.3 - n(r) = no[1-A(r/a) ]2 -

-l.0 1 1 1 1 1 1 1 1 1

0.0 0.2 0.4 0.6 0.8 1.0

normalized radial variable r/a

Figure 4.5 Comparison of TE02 eigenmode field distribution on

<>eprofile. graded-index fiber (10w contrast)
obtained by numerical solution of integral-operator
equations with first-order perturbation field
correction.

CHAPTER V
CONCLUSION

With conventional boundary-eigenvalue analysis for EM wave
propagation exact solutions exist only for planar-slab structures
or fibers having circular or elliptical cross-section shape due to
the separability of boundary conditions at the core/cladding interface.
Numerical or approximate-analytical methods are required for other
relatively-complex core configurations. Perturbation analysis based
upon an integral-operator description of surface-wave modes guided by
heterogeneous, open-boundary waveguides of arbitrary cross-section
shape avoids the problem of inseparability at the interface. To my
knowledge, this is a new unexplored analytic approach.

In the preceding sections, perturbation analysis based on the
integral-operator analysis of open-boundary dielectric waveguides has
been investigated. Excellent agreement with potentially-exact numerical
solutions have justified the approximations made for the perturbation
solutions for small perturbations in the dielectric core. For larger
perturbations, various inaccuracies are inherent in the perturbation
theory analysis, and the transcendental or approximate, closed-form
perturbation soltuions are no longer valid. First-order "corrected"
modal fields have been obtained and are again accurate for small
perturbations.

Implementation of the integral-operator perturbation method has

50

51

shown the adoptability of these farmulations to numerical and approximate
analytical solutions. The method can be used to study the characteristics

of arbitrarily-shaped waveguides with diverse graded-indices of refraction.

REFERENCES

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

REFERENCES

D. Gloge, "Some Lightwave Communication Studies", 1976 USNC/URSI
Meeting, University of Massachusetts, Amherst, Massachusetts,
Digest p. 122, October 1976.

E. Schutzman, "The National Science Foundation Program in Optical
Communication Systems", 1976 USNC/URSI Meeting, University of
Massachusetts, Amherst, Massachusetts, Digest p. 99, October 1976.

J.R. Whinnery, "Survey of Integrated Optics Work in the U.S.",
1976 USNC/URSI Meeting, University of Massachusetts, Amherst,
Massachusetts, Digest p. 99, October 1976.

R.C. Pate and E.F. Kuester, "Fundamental Propagation Modes on

a Dielectric Waveguide of Arbitrary Cross Section", Sci. Rpt.
No. 45, prepared for U.S. Army Research Office under Contract
No. DAA629-78-C-Ol73 by Electromagnetics Laboratory, University
of Colorado, Boulder, Colorado, February 1979.

D.R. Johnson and D.P. Nyquist, "Integral-Operator Analysis for
Dielectric Optical Waveguides -- Theory and Application", 1978
National Radio Science (USNC/URSI) Meeting, University of
Colorado, Boulder, Colorado, Digest p. 104, November 1978.

S.V. Hsn and D.P. Nyquist, "Integral-Operator Analysis of Coupled
Dielectric Optical Waveguide System -- Theory and Application",
1979 National Radio Science (USNC/URSI) Meeting, University of
Washington, Seattle, Washington, June 1979.

D.R. Johnson and D.P. Nyquist, "Numerical Solution of Integral-
Operator Equation for Natural Modes along Heterogeneous Optical
Waveguides", 1977 National Radio Science (USNC/URSI) Meeting,
University of Colorado, Boulder, Colorado, Digest p. 89, November
1979.

J. Mathews and R.L. Walder, Mathematical Methods of Physics,
New York: W.A. Benjamin, Inc., 1965, Chapter 10.

 

D. Marcuse, Theory_of Dielectric Optical Waveguides, New York:
Academic Press, 1974.

 

R.E. Collin, Field Theory_of Guided Waves, New York: McGraw-Hill,
1960, Chapter 11.

 

52

53

[ll] R.F. Harrington, Time-Harmonic Electromagnetic Fields, New York:
McGraw-Hill, 1961, pp. 125-127.