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Examination of optical fibre inhomogeneities and their effects on fibreExamination of optical fibre inhomogeneities and their effects on fibrecouplingcoupling

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Al-Jumailly, Tarik A.K.. 2019. “Examination of Optical Fibre Inhomogeneities and Their Effects on FibreCoupling”. figshare. https://hdl.handle.net/2134/13899.

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LOUGHBOROUGH. UNIVERSITY OF TECHNOLOGY

LIBRARY AUTHOR/FILING TITLE

___________ Ak;tll_~~'-'=-'=:t.:~_._T_A~-- ----~ __ .

------ ------------------~--- -..., ---------- --------...,...

)

....

EXAMINATION OF OPTICAL FIBRE INHOHOGENEITIES

AND

THEIR EFFECTS ON FIBRE COUPLING

by

TARIQ A.K. AL-JUMAILLY, B.Sc., H.Sc.

A Doctoral Thesis

submitted in partial fulfilment of the requirements

for the award of Doctor of Philosophy of the

Loughborough University of Technology

Hay 1984

Supervisor: C. Wilson, M.Sc., Ph.D.

Department of Electronic and Electrical Engineering

C Tariq A.K. Al-Jumailly, 1984

' Leush::,oro-v.;h Unlvtcrs"J

of T .,:::hno:o'Jt Ui;rtry

g;;~:;-~ -Cha••

!~ ~oSS 7 7--/o'1-'

To my wife Muntaha -------------------

(i)

CONTENTS

Acknowledgements

CHAPTER 1: INTRODUCTION

1.1

1.2

Review of optical fibres

Formulation of the problem

CHAPTER II: COUPLING THEORY

2.1

2.2

2.3

2.4

2.5

2.6

2. 7

2.8

Introduction

Elementary Analysis

Launching Efficiency

Application to LED sources

Inverse ~-power law profiles

Elliptical Fibres

Coupling Errors

Discussion

CHAPTER I II: INFLUENCE OF CENTRAL DIP IN THE REFRACTIVE

INDEX PROFILE ON THE LAUNCHING EFFICIENCY

OF GRADED INDEX OPTICAL FIBRES

3.1

3.2

3.3

Introduction

Theory

Discussion

CHAPTER IV: EFFECT OF CENTRAL DIP IN THE REFRACTIVE INDEX

PROFILE ON BIREFRINGENCE IN GRADED INDEX

page

2

3

3

4

6

6

7

14

21

30

37

45

64

66

66

68

83

SINGLE HODE FIBRES WITH ELLIPTICAL CROSS SECTION 85

4.1

4.2

4.3

Introduction

Theoretical Background

Discussion

85

86

95

(ii)

CHAPTER V: EFFECT OF THE CENTRAL DIP IN THE REFRACTIVE INDEX

PROFILE ON PULSE BROADENING OF ~illLTIMODE OPTICAL

FIBRES

5.1

5.2

5.3

Introduction

Theoretical review

Discussion

CHAPTER VI: EXPERIMENTS

6.1 Fibre End Preparation

6. 2 Measurements

6.3 Discussion

CHAPTER VII: CONCLUSIONS

Suggestions for further studies

APPENDIX A

APPENDIX B

APPENDIX C

APPENDIX D

REFERENCES

97

97

101

118

119

120

124

132

133

136

139

142

144

146

154

- l -

ABSTRACT

Coupling effects from source to optical fibre or from fibre to fibre

are very important in the design of communication systems using such

fibres, since both the amount of guided power and the output pulse

shape depend on them.

This work is an investigation into such effects.

Various fibre imhomogeneitieshave been studied analytically. Internally,

these are the refractive index difference A, the core ellipticity and the

difference in numerical apertures; externally, the factors considered

are fibre separation, offset and tilt.

The problem is treated using geometrical optics and is applied to

various profiles of different refractive index exponent (a). Fibres

produced currently by the (MCVD) technique normally have a dip in the

refractive index profile at the centre of the core. This dip in the

refractive profile affects the launching efficiency and causes pulse

broadening in multimode graded index fibres and birefringence in single

mode graded index elliptical fibres.

parameters has been investigated.

The effect of the dip on these

Finally experiments have been carried out to test the validity of some

of the theoretical derivations.

- 2 -

ACKNOWLEDGEMENTS

The author wishes to express his gratitude to the following people

for their various contributions in the preparation of this work.

To Or C Wilson for his constant guidance and encouragement throughout

the period of this research.

To the Head of Department for permitting the research to be carried

out in the department and for providing the necessary equipment.

To the Military Technical College (MTC) in Baghdad for the award of

the Scholarship for the duration of the research project.

To the members of staff in the mechanical workshop of the Electronic

and Electrical Engineering Department for their assistance in the

mechanical fabrication of the various items used in the experimental

part of the project.

Finally I would like to record my gratitude to my dear wife and

family without whose moral support the work would not have been possible.

- J -

CHAPTER I

INTRODUCTION

l.l Review of Optical Fibres

The advent of high quality optical fibres added new dimensions to the

field of communications. A great deal of attention is being devoted

to this technology since bandwidths approaching GHZ, and attenuations

. (1 2 3) below 1 dB/km have been ach1eved ' '

The transfer of signal from the source to the fibre, and from fibre to

fibre involves fine geometrical and mechanical accuracy. Therefore

research has been oriented towards reducing coupling losses and achieving

acceptable coupling efficiencies.

An optical fibre waveguide consists of two concentric cylinders, the

inner known as the core, and the outer known as the cladding. If the

refractive index of the core is higher than that of the cladding, light

is guided along the waveguide core, while the cladding provides protection

against oxidation and mechanical stresses. Optical fibres are classified

into two groups according to their refractive index profile: step index

and graded index.

After light has been launched into the fibre, it is collected and carried

by the fibre in the form of modes. The ability of the fibre to collect

as much light as possible from the source is a function of the launching

conditions. Modes that propagate in the fibre do so with different

velocities, hence the refractive index in the core is graded in such a

way that mode delay differences are equalized.

- 4 -

h f d . . . f.b . d 1 d . d 1( 4 ) T e sources o 1spers1on 1n 1 res are 1ntermo a an 1ntramo a .

The first part vanishes if mode delay differences are equalised. The

second part represents an average of the pulse broadening within each

mode and is the only term that is experienced in single mode fibres.

Intramodal dispersion arises from two distinct causes.

1. Material dispersion This arises from the fact that the light

sources do not radiate at a single frequency but comprise a

finite spectrum of frequencies and because the refractive index

of the material is frequency dependent, it ascribes to each

frequency component a different velocity of propagation.

2. Waveguide dispersion This is due to the dispersive nature of

the types of waveguides to which optical fibres belong. Compared

with step index, graded index fibres have a much higher bandwidth

and are less stringent to alignment requirements.

1.2 Formulation of the problem

Efficient power coupling between source and fibre or fibre and fibre

represents a fundamental problem in optical communications, for both

the amount of guided power and the output pulse shape depend on it.

The solid angle at which light is emitted from the end face of a fibre

is determined by the numerical aperture of the fibre, the launching angle

of the source side and the degree of homogeneity in the fibre.

Efficient coupling is a function of intrinsic parameters and extrinsic

parameters. h . . . ( 5)

T e 1ntr1ns1c parameters are : maximum index of refraction

difference between the refractive indices of the core and the cladding (~),

the index of refraction profile parameter (a), the radius and the core

ellipticity. h . . ( 6)

T e extr1ns1c parameters are separation, displacement

and misalignment.

- 5 -

As light is launched into the fibre, refracting rays would be extracted

immediately leaving only guided ·and leaky rays. Leaky rays would

gradually radiate from the cladding at a rate depending on their skewness.

The output at the other end of the fibre will be a combination of trapped

and leaky rays. For multimode step index fibres these rays can be

identified on the far field radiation pattern as inner and outer regions

respectively. The amount of optical power accepted by the fibre depends,

ultimately, for a source of given radiance and size and for a fibre of

given numerical aperture and size, on their coupling condition. The

best coupling of power occurs if the two surfaces are coaxially joined

together. Such a configuration also gives the best coupling even if

the source area is smaller than the core area( 7). Any deviation from

this condition, whether intrinsic or extrinsic, causes a loss in the

coupled power. The effects the various parameters have on the coupled

power have been investigated in some detail.

Fibres produced currently by the chemical vapour decomposition (CVD)

technique, normally have a dip in the profile(S) at the centre of the

core. This is due to the out-diffusion of doping material during the

collapsing process owing to the evaporation of the dopant. The shape

of this dip has been assumed to be gaussian(S) and its effect on propa­

gation characteristics has been investigated( 9).

It is however possible to assume that this dip can have the shape of an

. f'l (10) 1nverse a-power pro 1 e . An investigation has been made to determine

how such a dip affects the launching efficiency, the coupling loss and

pu~se broadening in multimode optical fibres and birefringence in single

mode fibres of elliptical cross section.

- 6 -

CHAPTER 2

COUPLING THEORY

2.1 Introduction

The electromagnetic theory of launching energy into optical fibres is

based upon Maxwell's equations. However, the mathematical complexity

(1) of the resulting description, led to an asymptotic theory , the main

. (2 3) component of which is the geometr1cal theory ' , which describes ray

(4) trajectories or paths of energy flow . The validity of geometric

optical analysis applies only when the dimensions involved are much larger

than the wavelength(S). If the dimensions are of the same order as a

wavelength, ray analysis can no longer be used to describe the propagation

in fibres adequately, and mode theory must be used.

In the following analysis, the spatial coherence characteristics of the

emitted radiation are neglected, because in geometrical theory, such

. h 1" 1 . ( 6) propert1es ave 1tt e mean1ng .

- 7 -

2.2 Elementary Analysis

A general ray striking the core-cladding interface is characterized by

its angle of incidence 9 and the angle 9 which the projection of the o n

incident ray makes with the tangent to the circular cross-section as shown

in Figure 2 .1. The angle e represents the inclination of the incident n

ray to the normal QP in the plane of the cross-section.

related to 9 and ~ by the following equation{?)

sinS sin~ = cos e n

Meridional rays are defined as those rays having ~ =

This angle is

(2.1)

11 z· Otherwise, rays

are defined as skew rays and ~ becomes a measure of their skewness. The

behaviour of rays in the fibre depends upon the angle of incidence, and the

critical angle may

e c

be expressed as:

-1 n2 = cos

nl (2.2)

where n 1 and n2 denote the refractive index values for the core and cladding

respectively.

' In the simplest case when ~ = 11/2, a ray will be trapped inside the core

providing that

and when

With skew rays

e < e c

e > e c the rays will be refracted;-

(2.3)

(2.4)

9 < 0 represents trapped rays, but for total refraction c

e > e and c

e > n

11 2

e c ( 2. 5)

and the ray is partially trapped, i.e. leaky.

- 8 -

TI When 6 > e and 6 <-- ec the ray is completely refracted.

c n 2

The numerical apertures (N.A.) for various fibres are as follows:-

For step index fibres:

where 0 = -1 cos

r 0

(-a

the core centre.

For graded index fibres

N.A. = [nl2(ro)

[ :>' ':"' N.A. =

1 - {_£)2 a

for skew rays

cos$ ) + $ , and r0

is theradial distance from

- n2 2p for meridional rays

2 ~ n2

for all rays 2 cos $

The fibre is assumed to have a profile of the form

r ~ a

= r > a

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.10)

The graphs in Figures 2.2 and 2.3 show the variation of the numerical

aperture as a function of the displacement (!.) and the projection angle~· a

The error (E) made when only meridional rays are considered to be accepted

by the fibre is:

E = N.A.M+S N.')!

(2.11) N.'\t+s

\

- q -

claddin core

'~----------------------------

a} RAY ON FIBRE FRONT FACE

I z

I I I ----L

I" . " I \

0 I core ......... I

:---.._ .................. -.t:1 / cladding (/) ......... . .--,, ---

fl ,,

b) CROSS SECTIONAL VIEW OF THE INCIDENT RAY AT THE BOUNDARY

Fig. (2.1)

Substitution gives

E =

- trl -

2 .l ~ cos ~

and this is independent of the index profile parameter (a).

(2.12)

Fig. (2.4)

shows the variation of this error with the displacement (~) for various a

values of ~. As anticipated, the correction factor approaches zero as

1T ~ ~ 2 which is the region for meridional rays only.

I t I i I '

-~~ g I 11 /

0 I

j: 1 l I t I : t ! I ' ! I

+ I

I '?

i I u fll I I _-1 I I

:_jl I I !

; I I

+<r N ~ : ' ' !

:' !

I I I· "! ;·

~: '[I !r'

~~ (

._;;

l /

/ 1 !

I

I I

I

/

G.J

/

/

0 '>D I

11 /

-e;-

/ I ;

0

"' 11

-e-

/

- 11 -

.. -~··· .. -

/ /

Numerical Aperture (N.A.)

1·-D

I i I

" <lJ

"" "' " bO

" 0

""' " 0 .... '-'

"' .... " "' >

-< z

z c ~ '"1

n

" ~ )>

" "' '"1

"" c '"1

"' ~

z )>

~

"C ~<ccc::;::-~~~------~~-~---"-a 0

\ ........ --------.:-,"-"-r- ~.-

\ '

' r r,

s~~-

C·~"> -- '

··•r,

r c. C• _)

' \ '

\ \

\

\

~~

\ \

\

o.3 ~-

0·')

I/

-----

'<> o· /

//

/ I

I

/ /

I

!

/ I

-="'

/' /

/

( i';l ~.;~:·_

-------------------------------------~--------------~---- .. ----- ---~----~ ------------------- -.-. '

-, 1 6 p ·J l3 ") i -1

Dif. (2.3) Variation of the N.A. with the oblique angle ~ for different displacements

~s 1

•/ 1 Q. \. I <..J

0

... 4>

~

N

"' "' "' 0

"' "'

'· ··, {. I

(_

f; _'

r)-

,_ ~

_,

) -'

~· L.

/

/

r

I

I

l /

I I

I 1

tpeoo

1k' b F q>- . - / /

/ z

rf) -n: I;:; 3 _,

--- ~ : ==:;0 "• . .~/ ' .. ---------~-- , ~rr -----..:..:-----------~------ ' ~;:~ 8 -- -~:- ----------, ,, . __ ,

Fig. (2.4)

-~ ') ,)

,, ,) F~ -_;

Variation of the error with the displacement (~ ) and the projection angle ~ a

~I ~ ..-, • c)

>: 1 s-. r ->(-) a

.... w

- 14

2.3 Launching Efficiency

An important quantity that is used to evaluate the quality of source-

fibre coupling is the launching efficiency n. It is defined as the

ratio of the power guided by the fibre P0

to the whole power emitted

by the source P.

Consider a circular emitting surface Se of radius b facing a fibre

surface Sf of radius a. Fig (2.5) with the source very close to the

f 'b . f d h f'll d . h . d h' 1' 'd{7) 1 re 1nput sur ace, an t e gap 1 e Wlt an 1n ex mate 1ng lqUJ.

of refractive index n', then following the analysis carried out in

reference 8 it is possible to determine analytic expressions for the

total power emitted by the source and the total power accepted and

guided by the fibre.

The radiance(q) of the source in the medium of refractive index n' is R'

(r', w', e', ~·)and is a function of the position polar coordinates

(r', w') of an emitting point on the source and is a function of the

angular polar coordinates (8', ~·)of a ray emitt~d by that point.

e' is the' angle the ray forms with the source axis. ~· is the azimuthal

angle with respect to the radius vector.

Since the two surfaces are concentric and very close then; r' ~

p =

·•· e' = ~o' e ' 0 • ~· = ~ .

0

. ( 10) The total power emitted by the source 1s

t r' dr' 0 J

2n J2n 0 dt/J' 0 d~' R'(r', tJJ', 8 1

, $') sin A'cose' dO'

(2.13)

w l5 -

w er en u. <(

D

~ w LJ er ::::> 0 Vl

!I: L:J _J

:z:: w w 3 f­w aJ

[:;: f­w L: Ea L:J

L:J :z:: _J

Cl. ::::> 0 u

-16

and the total power accepted and guided by the fibre is

p = 0

R(r , w , e , ~ ) 0 0 0 0

x T(r ,8') sinS' cos8' d8' 0 0 0 0 0 (2.14)

. ( ll) where a' is the minimum between a and band T(r , e') is the transm1ttance

0 0

at the interface between the fibre and the medium of refractive index n'

. ( 12) and is g1ven by :

4n' n i.e. T(r e') T

0 (2.15) = = o' 0 0 (n' n )2 +

0

For sources whose radiance is given by

R • < e • ) = R cos ~e' 0

(2.16)

~ = 0 represents a Lambertion distribution and as ~ increases the

radiation pattern becomes narrower and finally becomes unidirectional

as R. ~ eo • From the definition of the launching efficiency given

previously, and for an (LED) whose radiation pattern is given by:

R' ( e') = R 0

the following launching efficiencies are obtained:

n =

for all rays (bound and skew)

and

n 2 2

2 cos ~

r dr 0 0

(2.17)

(2.18)

n' =

for bound rays only.

- 17 -

r dr 0 0

For the proof of these equations see Appendix A.

For a source whose radiation pattern is given by:

R'(S') = R cos ~8' 0

The launching efficiencies are(lO)

~ > 0

= Ja' J2" 1-~·1 - _n_( r-"-~ _> _-....,n-~ -]9, ; 2 r

ro2 2 o o o . n'2(l _ 2 cos ~

dr 0

for all rays (bound+ skew)

and n = 2 T

0

for bound rays only.

a

r dr 0 0

The variations of the launching efficiencies n & n' with the refractive

index profile (a) are shown in Figures (2.6) and (2.7), withthe source

directivity 't' as a parameter. Both nand n' increase with increasing

·~·, i.e. more directional source. Also both of them increase with

increasing (a); i.e. larger angular acceptance.

Fig. (2.8) shows a comparison between the two launching efficiencies.

(2.19)

(2.20)

(2.21)

(2.22)

- l8 -

CJ ":; ·1' Cl \ 'i r( l r· ('/ t ' - r! V>: I

11 ' ~i I I ~I

\I i

~\ I <>-< ~\ "\ I I I Cl

\ -<~ I

\ -c~ I fw )<

i ( J I I

~ \. + I I I

\ I \

I \ I I UJ I \

\ ··c·-J

\ \ \ 1 ~

1 t " \ ~

' ' I I< I

\ ~~-- Q)

I

I '-' \ \

('J Q)

I e I \ "' I<

"' \ I i I a. I

\ \ ~ 1

i Q)

t i ..... ....

\ I ('J "" \ ·(I 0

I< I I a. \ I I i \ I " \ 'Q)

i "0 I I I I " I, ' I ....

I

~ I i I I

Q) i \ t ID "' :\: l r ·I '-' \ I

I "' ' \ ..... \

\ I ....

\ \ :>

\ ~.

\ "' I I ~ I

\ I I 0) ,.,

\ I

\ u

' " :I: \ Q) .... I I t)

' .... I I "" i ' "" \ I I "' I UJ bO

\ \ i .. ~ " i

J ....

\ \ I "' \ t)

I

I " \ ,j; " :\; "' ,..., \ I

\ '"'

I

' "" \

' I 0

\ \ I ·-· " \ I

\ 0

\ I I .... \ i '-' \

\ i "' \ ' i .... \ \

I I< \ I i "' I

.\ \ cl; ! !> \ ! ; r...: \ ' '

\ ~- .- . ~

\ \ \ ! "' \ \

I "' ' ~

I bO

( I .... "" r. c ,l .. j_ '" . . .;. r···-· - ----· -- - ---· --- --- - -~ -- ------ ------··-··. -·

,_::-) w r- lC ..!) ~- ('/ " ::~1

Launching Efficiency

r " "' :l n ::r ,... " "' "' ,..., '" ,.._ () , .. "' " " ·<

··rr, I <- .,

r-. ;:_ -

r ~- r:,

r~ ,..)_ ! ....... t. -·

.-1 r:._ 1

·~

/\.D I ' o;_· -! .

·f· -") (~ ..J ' . -" . """"'r·~ ; ~..__. ;~-,.

"""""\(:_

·-~ --~----------.._.

~.-,

..:.. (...<-I

________ .-e--· ·b-----

~-

-----~-~---------

------~ ---Cl

______ -e---- .--e-----­--

--.---r--

--- ... __...._'

--------------------'?'>------------~-

~---­-------~

------~-:.

L~o -----+------------+

-----+--------------+------------+-------+---+--

. ------· ------r----------r:.:, I-·-·

;- ----- --------~-----------,------- -------.---- ----. ---------- ---- -----------, 13 1 2 ]A lG 1 8 25 :::8 38

Fig. (2.7) Variation of Launching Effiency n' with the index profile parameter (a) >' 1 .-;~-1 ' I u -+(a)

,... "' " " ,... "' "'

r.,:·;. -· .. _

. ·: r-, 0 ~-· RO

.-10 ·~

- .• ( t ~J {.'

'""'C ..

. '

+

, . .) - -·

.

__ ...;.r:· .----

--.. ....-----.......... -

--. I -·~·-

--- V\ -·?·-

----~<

----------;:_:~ 'L V\. --"--------. ________ ...x--___________ .

..... .;..-:-.. -·

---_.:r---------

--~----.. ---··

______ ,x----------------- --- --'+·---­

----

_ __.,._---------~------·---_.:,.-----

'L: .. ·t I ... -·-----;'

_____ ..... _____...,.-----"\ ... -- -_,....- ........ ----____ __,._... _______ _

--A-------------~)

.. ------ ______ .--() ___ ,.a--------------..-e--··-·-·

-------------Ft-·---·--------e---· Y\ ___ --f'r-- --------- \.- ..,. 0

... - ----+-- ---- ------+---- -----+---..

---- ...,------------+--- -T:--- ------- ~--------- / 1- - (J ---- - \\

--+·----·-

..... -- ., ........ _ .. ________ -.... ·---:-· -- ----------r--··--------···-.. -~----·-------- --. ___ .. _ ----·------·----r·-·----------·-:-·-------·-r----------, ..., L 1S 18 22 2~ 26 28 38

-+(a) Fig. (2.8) Comparison between the launching efficiencies n ~ n'

- 21 -

2.4 Application to LED Sources

For an LED source, the launching efficiencies are given by equations

(2.18) and (2.19) of the previous section.

Applying these results to fibres having power law profiles of the form

for r ;S a

and for r > a

analytical expressions were obtained relating the launching efficiency

to various source and fibre parameters.

Consider first the case of bound rays only, i.e. equation (2.19).

Here, two alternatives are to be considered

i) b ~ a (source radius larger than the fibre radius)

In this case the launching efficiency is given by:

n' ==

ii) b < a

4 T n 2t. 0 0

,2 n

1 Cl + 2

In this case, direct coupling does not give the maximum possible

(2.23)

(2.24)

power input to the fibre since the light does not fill the numerical

aperture cone.

r

l n2ll I

b Cl 4 T l i 1 (2.25) ' 0 0 -- (-) n

n'2 Cl + 2 a .I

and these two expressions match when b a.

- 22 -

For the proof of equations 2.24 and 2.25 see Appendix B.

These results were plotted in Fig (2.9). It can be seen from the graph

that as the index profile parameter a increases, the launching efficiency

increases for given values of b/a and approaches that of the step index

fibre as ex + oo

Secondly, when considering all rays (bound + skew) the launching efficiency

for the above two conditions is given by:

i) b ~ a

n

ii) b < a

n =

2 T 0

n 2[.. 0

2 T n 211

0 0

2 1fa

These results are plotted in Fig. (2.10).

rdr d<j>

I rdr d<j>

Figure (2.11) shows a comparison between the two launching efficiencies.

(2.26)

(2.27)

For the region where b > a, the launching efficiency decreases as b increases

due to the fact that the fibre is not capturing all the light that is

emitted from the source. For b < a, the launching efficiency increases

as b decreases, due to the fact that the angular acceptance of the fibre

is maximum at the centre and decreases towards the edge of the fibre.

Higher launching efficiencies are obtained for higher values of the index

profile parameter (a) and reach a maximum value for a-> oo (step index fibre).

{'\j-

.. ,,··

') .-, ·' (_'.'

n' -~, ~­_;

r " = ,..., .-,

" •. ()

=:-.... " "' "' "' "' .... _,

' () ,... (1)

" ()

'<

·-:-

r. ..._, __

Fig. (2.9)

-- - - ·- ·- - ·----.. \

';.,• \

' \

-- -------- ________________ .. _____ ,, .... _.. ____ -----~--- ---· ....... ___ , __ ...

F.:i

\ \ \

•0 lu

\ \

\

\ \.

\

\ '\

\.

.....

'""''"~ --·---~-~----·-·- --------·- ------- - ... ·-· ------;-· ---·-----------------.---------, 16 :s ~8 , ')

' ,_ 1 .-·l

Launching Efficiency V.(~) a for different index profiles (a) For bound rays only

-, (_'.

-.c ' --'

n

' r " c ::J (> ,. ~-:1

"" '-

"' ..., "" ~· (>

~-ro ::J (>

'<

.--. -,

Fig. (2.10)

5 ''> i '-·'

Launching efficiency (bound+ skew) V.(~) a

' ') ; '-- ' ' ! . -

for differenta values

------- -- -..~-. __ -....;___ __ _

:e ._, 1 --~. --;". J ,_;

---.....: o<"' I

--, -.. ...... ·

b -+(-) a

.. ,.

'Y)

'<::l .... ><

.... ~ c.

n & n' .. ., " " 0 :T ,. "J • :~ .... .. '·

" (IQ

"' .., "' .... 0 .... 0

"' " 0 '<

... ......_

Fig. (2.11) Launching efficiencies of n &

'· \

\

\;

< ., ; /_ 'C I ·J

n'versus (~) for different values of the profile index ~ a

·-- ~--"-·-

1 c-: '~

·; .--. . - ~.;

/: 3''

-26

Also an improvement in the launching efficiency is obtained when the

effects of skew rays are taken into account.

Coupling loss, which is defined as L = -10 log n is shown in Fig. (2.12) c

and(2.13).

Fig. (2.14) shows a comparison between coupling losses for the two types

of launching efficiencies. Lower coupling loss is obtained when the

contribution of skew rays is being taken into consideration. This is

.due to the increase in the value of n for the bound+ skew ray case.

() 0 c:

"' -,... ::1

OQ

r 0

"' "' r

n

~

0.

"' ~

- ., ~· ,.) .

-, "-

""'lO "--'·'

(

C• '·-·

r· D

• t : ~-

·---·----· : ..•. ------·· -------------..------ .. -------. --------------··----- -------------------------------- -- ------.----------------- -----...,

Fig. (2.12)

-\ ,, .J

o C)

i ..:. ~ '7 l' :s 18 ~0

Coupling Loss V.(£} for different a values for bound rays only a

b +(-) a

(") 0 c ., ,..

-0

"' "'

.. ,

) ")

·- J

') .

'l' ·- '

")I? ' ·~ -:.-.'

' r, I t'

.,

• c .. _)

Fig.

....... 0. = 1

2

2.5

3

~ ~-- --~-------- ----- -- ----------- ---:----------------------, ---------- -----· ----- ----· -------~--------·- ----, ,, 4 5 s I r~

I~· 1 2 1 ,1 15 :s

(2.13) Coupling Loss V.(~) for bound + skew :<1~;-·

:t rays

------------------------

N 00

~; . .., ,' ~ ,_

l i

" ~ ! . ·' I

( ' ·'.

' ' ' ..... :

.)

Fig. (2.14) Comparison between

--·----r• 'J

-------

/ /

'0 lu

/

/

~ ') I .;,_

coupling losses v. (E.) a

for va · r~ous

.1 i()

ex values

:e '' 1 n-, t\ : .::.) ;

l

--., . .,0 -- ..... _r

- 30 -

2.5 Inverse a-power law profiles

Inverse a-power law profiles( 13 •14 •15 ) have a refractive index

distribution of the form:

2 2 II n(r) = n -0

2.s0 - ;)a for o < r ~ a

and • (2.28)

2 2 [1 nJ 2 for r n(r) = n - = n2 ~ a

0

as shown in Fig. (2.15b) where all the parameters are as defined

previously and o is the relative refractive difference between the edge

and the centre of the core. It can be considered as a measure of the

dip height(l 3).

Using equation (2.19 and following the same reasoning as explained in

section 2.4, the launching efficiency for bound rays is given by:

i) b ~ a

4 2 n2 To

(!!.)2 [ 6 _ .s[.!. _ ~+ a(a - 1) } l n = ,2 b 2 2 3 8 n

(2.29)

ii) b < a

4 n 2 T

[% -.s [~ - (~) a(a -1). (~)2 }] n' 0 0 a

3 + ,2 a 8 n (2.30)

- ) l -

n(r)

r=a r

_.!!1 Cl..- POWER LAW PROFILES

n(r)

r= a r

b) INVERSE t:f-_- POWER LAW PROFILES Fig. (2·15) POWER LAW PROFILE FIBRES

- 32 -

From equations 2.29 and 2.30 graphs of n' against b/a are plotted for

various values of 6. These are given in Figures 2.16 and 2.17.

For the region b > a, these fibres behave exactly the same as power

law fibres. For the region b < a, the launching efficiency is a minimum

at the centre and increases towards the core-cladding interface. This is

due to the angular acceptance of such fibres increasing to a maximum value

at the core-cladding interface.

It is evident from the graphs that the smaller the value of 0 , the higher

the launching efficiency and the maximum launching efficiency is obtained

when o is zero (step index fibre).

For a fixed value of 0 , the launching efficiency improves greatly as the

profile parameter a is increased, so higher values of 0 which tend to

reduce the launching efficiency could be compensated for by an increase in

the profile parameter a·

Unlike power law profile fibres, which have maximum launching efficiency at

the centre and when the ratio of the source area to the fibre area is minimum

these inverse profile law fibres have maximum launching efficiency at the

core-cladding interface when the source area is equal to the cross sectional

area of the fibre.

Coupling losses were plotted in Figures (2.18) and (2.19) for different

values of the index profile parameter a· From the graphs it is clear that

these fibres have a maximum coupling efficiency which is less than

that of tl1e power law fibres, ~tJt because of tl1e nat11re of tl1cir

angular acceptance, these fibres can best be operated by light sources whose

surface areas are as large as the cross-sectional area of the fibre.

is not the case with power law fibres.

This

-·'

--· /

.··-' '·-'

/.

·~

0 '-'·'

/

' <l

' \_ ..

./ f ___...x-··

.. /.·------- +-. e

a.= 2 ~----------_A<-•'

··-r'

a = l

~-------.:,.;:---

. _ . ..;-.-. _ .....

-*----------~

... ;r \ ,..

0 = 0.002235 + e and f

0 = 0.004465 + c and d a = 2 0 = 0.0066975+ a and b

d ·/-1 _/>//

:;

/ v:- a

Fig. (2.16)

./

/

.... ,. --·-···· -·-······-------·-···· ··--· ------- .... - ---·---.............. .

' --~

'" -·-···--···---··-- -·········-·-··-·- ·-----·-·· ..... ___ -. --·-···---. ')

1 -

Launching Efficiency V.(~) for different dip heights (o) and different i.ndex a profile parameter (a).

- . ---' -· _,_,. --- ···---., -, •") . ~' . '-' . - '···

(~) a

'· ' I ·'"> ·• ' ' ' ' ' I ~..-

'·'

c-

>. ''_;, .. u c ~

u -•. .... 4-< 4-<

"' 0/}"

c .... u c " "' ""

)

0

f

/

0

'

I

Fig. (2.11)

a = 3 _.;...:-·· --__ _.....

a = 2

--·-···---- ----------

profile parameters (a) a

; '-'

\ ·- .,

\

••••• -- --- ------- - ···- ·------- '•4 .... -·-·

a a 0

= 0. 002235 + e and f

= 0, 004465 + c and d

= 0.0066975+a and b

• r: ' ' l ·-' u

'I ' •'').

Launching Efficiency V.(~) for different dip height (o) and different index

. -·-·······-······-····-------------------~---------------= ,\ I . ..'

~

"' '0 ~

.. :J ~ ~ 0 ~

00 c -~ ~

Q. ~ 0 u

·,.

::~· I '

I I

,. ! u.l

'

,Q /

r. ,_, .

. ~ .)

n /,

· . .,_

Fig. (2.18)

6 ~ i 3 1 .1

Coupling loss V.(E.) for different dip heights ( 0 ) and different index profile parameter~ (Cl)

./·a

b

0 = 0.002235 -+ e and f

0 = 0.004465 -+ c and d

0 = 0.0066975-+ a and b

-----------·------, 16 :s ~3

b (-)-+ a

~

"' "C ~

u ,_.,

"' "' 0 .... till ~ .... ~

p. ~ 0 u

~. '.""J:.

i --,)~7~ ~ _,.

i i \' I

'c : .• I

·.-1

'-·~ ·-

> '":·! ... ,

! --

; ~~-!

b':l

(. (~ '~'.,

· .. ..)

I ' ' i i

Q

•• '

\ .

\ 6

;s_

d

·.a

' '· \

~,

'-

\

Fig. (2.19)··

l

·coupling loss V.(£.) for pro file parameter a (a)

r' /

0 :

0 :

0 :

w 0.002235 ~ ~ and f ~

0.004465 ~ c and d

0. 0066'n5 ~ a and b

. ··--··--··-----:-·------------·-·--------,

!. I G ~ 8 ~8

( b) '<'•"•' - -+ - 'i a , , .... different dip heights (o) and different index

- 37 -

2.6 Elliptical fibres

In the previous analysis, it has been assumed that the refractive index

profile within the core is of the form n = n(r), which means that the

contours of constant index are circles. In practice, however, the

fibre manufacturing process often introduces geometrical or dielectric imper-

. (16) fect1ons . Barrel and Pask 07 ) have shown that fibres need not have

circular symmetry in order to retain the desirable properties of the

circular power law fibres, since the power accepted and guided by the

fibre is a function of the degree of homogeneity of the refractive index

exponent (a) and not of the shape parameter. (18)

Petermann , has shown that

the introduction of a small ellipticity in the index profile, leads to a

considerable change of leaky mode behaviour in the case of square-law

fibres. Ramskov and Adams(lg) have shown that the presence of small

ellipticity in parabolic index fibres (a ~ 2) leads to a large increase

in the minimum Rttenuation of leaky modes. This can be of importance

for coupling purposes and could lead to an improvement in the launching

efficiency of optical fibres.

In polar co-ordinates (r,e), the refractive index profile, for parabolic

index fibres (a= 2), can be written in the form( 2l):

2 2 [l

2 2 - 2 1 r (1 (2.31) n (r,6) = n - 2/; - e cos e)

0 2 y

where e is the ellipticity which is given hy

e2 2

= l - L 2

X

x and y are the major and minor axis of the ellipse respectively.

I ..., ''I 'o ~

)<

J !/)-

25

,.... i " c I

::J 20~ I">

::;"

::J i

"" I "' I ,.., ,..,

i 5 j .... I"> .... l " I ;;J I"> I '<

i " " . 9 I ;:; I _ .... ' 0.

"

-1 ) J

I I 0 -· ' 0 2 4

Fig. (2.20)

e = 6%

6 8 10 1 2 1 4

Variation of the launching efficiency as a function of (~) different ellipticities for parabolic graded index fibre:

16

for

20 b -+H a

2 ~;). L ')( ib ·

t ')"'

30

25

22

·. c; I··*'

10

5

' <:" ~,..

"' >l<

s" '

' 0

' '

<l'~

Leaky mode reg~on

N

"'

0

N

" 11

<0.

2fa 2 " + 2 12 -B 2--

( ....?..) (no 2) " =

max 2 " + y a .,

0

--· 0 ·- ·o

Bound mode region

0.:---~.----~----.,----.-----.----,----,-----,-~-,----,-----,----r----.-----.-~~ ·--._

0

2080 2085 2090 2095 2100 2105 2110 2115 2120 2125 2130 2135 2140 2145 2150 2155

Fig. (2.21) Ray domains (B - L) for parabolic graded index fibre X10-3 '2 _. B

- 40 -

Following the analysis carried out in Section (2.3) for circular fibres,

the launching efficiency, taking into consideration bound rays only,

was calculated for different values of e.

Fig. 2.20 shows that the launching efficiency improves with increasing

ellipticity. .(22) It has been suggested by P.F. CheccaccL that the

number of bound modes increases with the introduction of a small ellipticity

in the fibre core. This can be explained from fig (2.21) which shows

the ray domains B-L for graded index circularly symmetric fibres (o ~ 2).

. ( 19) In elliptical co-ordinates (c,A) the refractive index profile is gLven by

2 n (£,A)

2 g(c) + h(A) ~n-.2 .2

o sLnh c + sLn A

where n0

is the index at the core centre,

and

g(e) ~

h(A) =

(n 2 0

2 2f 0 .2 o (n - n X-) SLn A cos A o 2 r

(2.32)

f' f is the distance of the focus from the centre of the core and

~ e r

r is the distance from the core centre to the core cladding

interface along the x-axis.

For the parabolic index elliptical fibre (o = 2)

g(e) (n 2 2 2 s inh 2 c cosh 2c = n2 ) e

0 (2.33)

and

h(A) (n 2 ? 2 . 2 2 . n2-) e sLn A cos A

0 (2.34)

- 41 -

The scal.iltwave equation for each mode E(x,y,z) is( 23 )

where n(x,y) is the refractive index and k =

space wavelength.

Taking E(x,y,z) = ~(x,y)e-jBz

where

In elliptical co-ordinates:

1 =

2 2 ( 0 h2 e r s1n e: + 0 2 ) s1n A

211 A

and A is the free

(2.35)

(2.36)

letting ~ = Q(£) S(A) and following the same analysis carried out in (23)

1

2 2( 0 h2 e r s1n e: + 0 2 ) s1n A

But 2 (£'A) is given by eq. (2.32) n

d2Q ? 1 1 d-s 2 2 (k2n 2 + - -- + e, r Q d£ 2 s dA 2 0

- k2e2r2

This equation can be separated into

and

d2

Q 2 + k G(£) Q(£) = 0

2 d£

+ 0

= 0 (2.37)

fg(£) + h(A)} = '"

0 (2.38)

(2.39)

( 2 0 40)

- 42 -

where

G(£) e 2r2 2 A2 2 - g(£)] (n 2 2 2 A = n - B )sinh - n2 ) r )l L 0 0

(2.41)

and

H(A) e·2r2 lno2 A~ 2 h (A) 1 (n 2 2 2 A

- 8 sin A - + n2 ) r )l 0

(2.42)

. . ( 24) where 8 and ll are the ray 1nvar1ants .

(16 23) It has been shown ' that ~liptical fibres support two types of

modes, hyperbolic or H-type and elliptic or E-type depending on whether

~ is greater or less than zero. In the H type, the mode is confined

to the regions:

0 < €: < £ tp

A • < A < n/2 m1n

and the E-type mode 1s confined to the region

o <A < n/ 2

It has been shown(l6) that the H-type mode can only be bound whereas

an E-·type mode can be bound or tunnelling.

Substitution ~f equation (2.33) into equation (2.41) and with the

mathematical manipulation shown in Appendix C gives:

• h2 1 2 A 2 2 s 111 £ = -- ( n - 8 ) - 1 +

tp e 2y2 o (n 2

0

A2 p -B) -1.1 (2.43)

- 43 -

and similarly

2 sinh2

£ • m1n

l 2 = 22 (no

e y (n

2 - ih

0

where

In' E-type modes, the 1nner and outer caustics are ellipses

(see Fig. 2 of Reference 16).

Let the distance from the centre of the core to the outer caustic be

r ' tp

.2 [ h2 e cos e: -tp

• 2 J s1n A • m1n

But the E-type mode is confined to the region where

A • = 0 m1n

=

= . h2 + s1n

Substitution of equation (2.46) into (2.43) gives

r + _l_ A2 .2 .2

l\ 21 2 (no 2 (..!E.) 2 2 (n 2 ~+ e = e - 8 ) -r

2Y2 0 2 2 l y e

,.

+ lf ·.2 r

(..!E.) 2 l { l (n 2 A 2) 2 l (n 2 §2)_ e<J - 8 + e -r 2 y2 0 y2 0

r

2 }2 s> - 1

(2.44)

(2.45)

(2.46)

(2.47)

(2.48)

__!!: r is a real quantity, therefore the term under the square root must

be positive, implying

l 2 2 A 2) 2 2; ~

Y (n - q - e > -o P e (2.49)

r ( tn) also -=­

r

- 44 -

should be less than unity, for if r is outside the core, tp

this type of mode is not possible. r must be within the core to tp · . 1 (n 2 ·2) ensure gu1ded propagatton, therefore for - 8 > 1 and to ensure

y2 0

that·

r ...!£. < 1

r

. (23) the following cond1ton holds :

From equations (2.49) and (2.50)

. •2 8

( n 2 - iJZ) + e2 = 2 0

= n 0

2

<.!!.... 2

.e

Equation (2.53) shows that the presence of ellipticity in_the fibre

core tends to shift the propagation constant towards the leaky'mode

region, in other words the presence of ellipticity reduces the ratio

of tunnelling rays to bound rays that the fibre can support.

(2.50)

(2.51)

(2.52)

(2.53)

- 45 -

2.7 Coupling Errors

Coupling errors are classified as follows (see Fig. 2.22);

l. Separation: when the source surface and the fibre surface have

the same axis but are separated by a gap (s),

2. Displacement: when the two surfaces have their axes parallel

and separated by a distance (u),

3. Misalignment: when the surfaces axes are at an angle (cr) to one another.

In the analysis carried out in Section 2.3 it was assumed that the fibre

and the source are concentric and very close. In the following analysis the

following assumptions were made:-

l. that the space between the source and the fibre is filled with a

medium which has the same refractive index as the fibre core so

that· power loss at the fibre surface due to reflection need not

be considered.

2. [' ' \ti..t. assumed,] \\.1+l't~~~~use of the geometrical optic multimode operation is

approach ..

The power emitted by the course is given by equation (2.l3)and the power

accepted and guided by the fibre is given by equation (2.14). The power

emitted by the source is not affected by the presence of any of the

coupling errors but the power accepted and guided by the fibre is, and

the existence of small gap (s) in Fig. 2.22 requires a transfer of eo-

ordinates from the source surface to the fibre input surface, on other

words, if the radiance distribution on the emitting surface is R'(r' ,Jll' ,e', q,'),

this should be transformed to the radiance at the fibre input surface which

- 46 -

s

u t

~-x--\a.: ___ _

1---- ---- L...-.----­'

Fig.(2.22.) REPRESENTATION OF COUPLING ERRORS

- 47 -

is R(r,$,8,~), this is done by relating each variable r 1, $1, 8 1, ~ 1 )

on the source surface to the variables (r,$, 8, $) on the fibre surface.

This was done in the same way as that carried out in reference (8) to

obtain

r' = r'(r o' $I = $I ( r

0, $ , 8 , $ ) , 8 I '= 8 I ( r , tj! , 8 , $ ) , 000 0000

$1 = $1(r o'

R( r , tj! , 8 , $ ) = R 1 ( r 1 ( r , tj! , 8 $ ) , 0 0 0 0 0 0 o, 0

•1 (r , $ , 8 , $ ),

0 0 0 0

8 1 ( r , $ , 8 , $ ) , $1 (r , $ , 8 , $ >]. 0000 0000

and if the interposed medium is lossy then R1 should be multiplied by

the transmittivity T0 (r ,8 1 ) which takes into account fresnel losses 0 0

at the fibre input surface. The mathematical transformation can be

found in (8,9) (no analytical formula could be obtained) and only

the results which were obtained by numerical techniques, are presented

here.

n'

::-' ~ c ~

" '"" ~.

~

"" '" .., ..,

" .... "' " " '<

) ") . ,(_ /. -i

' ,.,.

'8 : ~ -1

;

,. . .) -

,. ' ; ,..

: ,-1.

•\

., ..)

I

' )

.. -, ' •.: -.

s. '

'"·+--·-·. ··-·-··-- -+ .. .. . ----------+-------- -----+----.. ····----------.......__ --~

----------

------ .. -- 7-------------- ;----- ----------, ----------or-----·-----------,----------~--------~------2 J A 5 6 7 8

Fig. (2.23) Launching Efficiency v. separation for bound rays only

b = 1

a

---.... ------------- 0( =--' -----4-

-- ,. -- -----------.-----------------.------, ~I 18 ' ' ' I

s + (-)

a

n

() .... " ~ n '<

.-. · . ..::

. - '·' -

·-

,. . ;; ~

.. -:

..

.....::....._ ___ _ ------ ~-- --

-----~---- ----------;.;._ . -.............. .._

------~---.......

. ------e----------&--------E-)-._._

-----E-). . ----

+·· ... -+-. . ·--+-

. ·- -----+-.

. ·-. . ----~-----

~----~

-------+-...

b - = l a

.._ -------+-,._ ·-·-

·-------- -----------·------------. ------------------,--------------------,----------·------·----5 G '7

Fig. (2.24) Launching Efficiency versus separation for (bound and skew) rays

·1 r, , . _ _;

'""-· ..

---~~: 1

- --,

..,. (.'!.) a

b 1 =

n' a 2

""_; ') .. .) ·-·

--·-

···-~-~---······-·-···· ------------r-··----------··-;---------------~---------;-·---------... --------·-----;----2 3 .t 5 G '7 8

Fig. (2.25) Launching Efficiency v. separation for bound rays only

'··

. ------~-- ----------------· -·.

I:J ' . ' ..., .,

' ·-

vo 0

r " c ;:J ()

::T ,.,. ;:J

OQ

"'· "' "' ,.,. () ,-. et) ;:J ()

'<

. , ' . .) ..

' ' ~--

··,r· / .:..

f-. )(:)

·i

-,1 i

t .. ..: . .j

"")!), L. •• >:.·

:5

., ···..::-- ...

. ·- .... _~---.

b 1 =

a 2.

................................................. __________ . ____________________ ,_, ........ ------. -----... ------------2 3 ~ 5 6 7 8 ~

Fig. (2. 26) Coupling Efficiency v. separation for (bound + skew) rays

'" ' c> ' 1

L___ _________________________________ __

' ,..., '

->- (~) a

C""l 0 c: "0

()

~-

~J -·

·:; ,··~ ._) ~- .

--,~.

' ) . "- ·'

"") r~ ' •.. ~- ! ' ..

. ("", i L•.;

, .. :~

.. -'

. r, ._, . i ;

E: .

·>

-----"'-._ 't:;;..._

---··"llo' ... __

··~-

. ·::-:-.

~------·--7(----'*-

3 .j 5

Fig. (2.27) Comparison between Launching Efficiencies v. Separation

1,2,3 b 1 ~

a

4,5,6 b 1 ~

a 2

1,4 " ~ 1

2,5 " ~ 2

3,6 " ~ 3

·>

('

6

5

4

··. ···.:... 3

.. .:....:._

2

1

···---···--- ----- ·-- --··-·-----·--··-------·· -~ ' ..... lo.J • I

.. (~) a

V> N

-,,-~, L..,

~- ·- ·-'....,

. .., .. ·, :";. ·- ._. ··-

" C; t. ,_. ·--· -'

n ( .. ·' 0 ....... c

" "0 ..., .... " "" 0 r·t: r ·~ .o 0

0

"' "' r

(}

~ 0 (··-· ,_. ~-

0-

"'

,, .. ' -· 0

C9 \)· .. ""1· .• ;

I<.... "

~-------~------~

1 (' ~~ : . <.)\:; Oo . -- ...... - ····-r·---·-------:----------·------·'T-------,-----~----- ---··r-------'"7"---·------ --------···--,

0 2 3 .-t 5 5 7 8 9 13 11 12

Fig. (2.~) Coupling loss v. separation considering ~ound rays only

.··,

' '­t)

~I )(tO

~- +·-·

) .j I

I - r·;.. 11,

·'"

0

.. ··-. --- -....;.+----~-- -------+E--

'- ( -.

-----. .---­.- ----{~----

~-

----·

/ /

__ /_..--/

__ //// //

---<-;"'

X

= 1 2

Cl = 1

2

.. _. 2.)

X 3

••••-• •••••••·-·•·•·--··••-•-- ·-----•·•--•- ••·----•• •• •···- ······- ···-·-··r··-·-·--- ·-·······•·----•••••• ···-·-·---•--·• ·-·-•· .. -·••••---•••••••••••••• ·---, (_'• 3 r ·,

Fig. ( 2. 29) Coupling loss v. separation considering bound rays only

2 .--. ,,

-+(~) a

() 0 .;. -.... ;;:

"" r 0

" "

'2.00

1'90

175.

170

1(,0'

ISS.

0

---+ . -+-·· ----- --·· + ---·-· -+---

____.:,.------------ " ___ ...:.---

. ----:---------

------~--

( 2

-· .. -· ____ .... --·· ___ -"';:"

'. '

.+_.

(-.

.--··'

---- .;<

.---·

Fig. (2 .• 30) Coupling loss v. separation considering (bound+ skew) rays

a = 1

2

2.5

3

..0"

... . . ..X _.

__ .-------··

b = l

a

------------------ ---------- .. ----.--------, ..,

. '

; • l''' ;.:) .J

(") 0 c:

"" ,..-~·

" ()Q ~~- (";.

t"' '-' ~-

0

"' "' t"' c.

~

' 0. 5tj "' ~

-~-

0

__ . .--··

0

.X

' /

,X '

b l =

a 2

·· ·· ·- -~ ·:-· ·~--- -----~· ... .... _ .. 1 --·-·-·•----------r-------··-r·----------·r-- ··--·------- ··;-·--------·--:--------, ----·-·----.......... T··-----------------:-

; 2 i 3 ~ s s 7 e :J 18

Fig. (2.31) Cou~ling loss v. separation considering (bound+ skew) rays

,-

Cl = l

-*

2

./'' 2. 5

---------· ---, 12

+(!) a

"' "'

C"l 0 ~

" ~ .... ~

"' r 0

"' "' r

" ~ 0..

"' ~

•, ..

\ ... : ·;

r qo I

IS5"J I I i-

I 'SO

1'15

170

1(,5_

1{,0

I 55 .,

1"):0

I !f5· [)

1 '-fo: 0

---*"""-------~----· ··--~----··-·---·-; -----

·-----e----·-___,e,r-

____ -e __ _

,:-

... --:-·· __..:.- .

.... ... --- .-"

-~---- ---

:- ~ -·

.. ---

--·

.. ··

--·-··

l' 2' 3 b 0.5 - = a

4, 5' 6 b

l - = a

-··-- ------ ______ ,, ___________ -- ____ ,,_, .. --- --- ... _1

Fig. (2.32)

c ·-·

Coupling loss v. separation for (bound+ skew) rays for different ratio of(~) a

©

··,

26.

24

22

20

1 B. "' "' 0 1 6 " "' ... 0 -~ 1 4 ""' ""' "' bO

" 1 2 -~ ..c 0

" " 10 "' ,...,

8

6

4

2

0"1 0 2

Fig. (2.33)

b 1 =

a <>

' s = 1 a

4 6 8 10 12 14

Variation of the launching efficiency with displacement for different index profile parameter (a)

1 6 20

22

20

18

1 6 -"' >. 14 u c: Q) .... u ....

1 2 "" "" "' M c:

1 0 .... .c: u c: ~

"' ,.., 8

6

4

2

0., 0

~

2 4 6 8 10 12

£. = 1 a

~ = 1 a

14

Fig. (2. 34) Variation of the launching efficiency 11 ' with displacement for different index profile parameter (<l)

1 6 20

26

24 ---22 --..... b

"*-- - = 1 \ •--- a

' 20 '*.._

!. = 1 ' a <="

"" --!3--• "0

18 --s.... ' c

"' "' 'a.. <="

' >. 16 ~ " u

" " 'a.. ' ...

" u 1 4. ·~

~ ""' ""' "' ")<. to 1 2 ~ ¥

"' c

0 .... <:; ;>< c 10. " " '

' ,_,

8 Cl = '

6 Cl = 2 X

4 a· = 1 ""

">< 2. -<:

)( \(

)( 0., >' 0 2 4 6 8 10 12 1 4 16 18 20

Fig. (2.35) Comparison between the variation ·of the launching efficiencies n' and et' (E.)X10-1

a with displacement

26

24

22

20

1 8 <= :»16 <.J

" ., .... . ~ 1 4 ..... .....

"' oo12 " ....

,!: <.J

10 " ~ "' ....,

8

6

4

2

0'1 0

Fig. (2.36)

b 1 - =

a

s = 1 a

2 3 4 5 7

Variation of the launching efficiency n with misalignment for different index profile parame'ter (a)

9

22

b 1 - =

20 a

s 1 =

a 1 8

16 <= ,., u 1 4. " Q) ... u ...

4-4 4-4 1 2 "' bl) c: ...

.!:! 10 " c: ~

"' ,_, 8

6.

4

2

0; 0 2 3 4 5 6 9

Fig. (2.37) Variation of the launching efficiency n' with misalignment for different index profile parameter (ex) ·

26 b 1 - =

a

24 s 1 - =

a -----liE-22. -~--- ----B- ~ 20

~ <= -B- ' -B- -B- "" -g18 -s- -s-' "' -s.... " c:

'El.... ' ~16 '8 "' " Q)

' ....

' .~ 1 4 lSl "' .... ....

' "' ' ""12 lSl = 3 a-a w " ' .... .c:

lSl ' u

" 10 ~

"' ....,

8

' 6. <> = 2

4 a = l

2

0, 0 2 3 4 5 6 7 8 9

+ cro X101 Fig. (2.38) Comparison between the variation of the launching efficiencies nand n' with misalignment

- 64 -

2.8 Discussion

Simple analytical expressions have been obtained for calculating the

launching efficiency and coupling loss of power law optical fibres. A

similar approach has been used for inverse -a profile fibres.

The analysis shows that the launching efficiencies for both power law

profiles and inverse a- profiles approach that of the step index fibre

as the limiting case. Power law profile fibres are more efficient in

power coupling than inverse a- profile fibres. This is due to the

variation of the numerical aperture and the acceptance angle, for in

the power-law fibres they are a maximum at the centre and decrease''

gradually towards the periphery while in the case of inverse -a profile

fibres they are a function of the dip at the centre of the core.

Elliptical fibres, in which the contours of constant refractive index

are concentric ellipses have been considered. It has been shown ( 2S) that

ellipticity in single mode fibres establishes preferred fast and slow

axes of propagation and produces a retardance which depends on the degree

of ellipticity. In multimode optical fibres, the presence of small

ellipticity in the fibre core, improves the launching efficiency of

parabolic index (a = 2) fibres. This is due to the fact that a small

___ ellipticity tends to shift the propagation constant to any intermediate

value between B = Bmax and B = n cladding·(Fig.(2.2l). This represents a

reduction 1n the ratio of tunnelling to bound modes. This shows that ellipt-

icity can be employed to improve the launching efficiency of parabolic index

optical fibres. . (26)

But it has been shown by Schlosser that the

ellipticity should be less than 10% otherwise it will greatly reduce

the bandwidth.

' !

- 6s -

Since this improvement in the launching efficiency is significant only

in parabolic index fibres (a= 2), an optimum value of ellipticity

can be found that improves the launching efficiency without affecting the

fibre bandwidth. This is of practical importance since the values of a

in the region a= 2 are the values of main interest as the optimum

profile for minimum pulse dispersion is close to parabolic.

When coupling errors are introduced, the graphs show that the launching

efficiency improves for higher values of the index profile exponent (a)

and therefore the coupling loss is reduced. The graphs also show that

the launching efficiency is less sensitive to small separation values

than misalignment and displacement.

- 66 -

CHAPTER 3

INFLUENCE OF THE CENTRAL DIP IN THE REFRACTIVE INDEX PROFILE ON THE

LAUNCHING EFFICIENCY OF GRADED INDEX OPTICAL FIBRES

3.1 Introduction

The best optical fibres available to date are usually prepared by the

chemical vapour deposition (CVD) method (l- 3 ). However, one problem

in this process which has not been completely overcome is the volatili-

sation of the dopant from the innermost layers during the high temper-

ature collapse, thereby yielding a refractive index dip on the fibre

. (4,5) 8X1S . This effect is commonly observed in fibres having binary

glass cores in which one component is more volatile than the other.

G bl . (7) . ff b . . . . . am 1ng suggested that th1s e ect can e m1n1m1zed by 1ncreas1ng

the concentration of the more volatile component in the innermost

layers and adding a less volatile component to the last layer.

The effect of this dip on propagation characteristics and pulse broad-

. h b . . d .b 1 h ( 6-S) h . 1 . entng as een tnvesttgate y severa aut ors . In t e1.r ana ys1.s,

the shape of this central dip has always been assumed to be Gaussian.

It is equally possible, however, to assume that the central dip can

have an inverse n-power law profile, as shown in fig. (3.1). The

effect of this dip on the launching efficiency is investigated in the

present chapter.

•

6

n (r)

- - - :;:::. __ _,..,.... .... / .....

/'?.--' : I/ ~ I ~/ ::;~-' I

..L..___ I I I I

- 67 -

------1--------)-----

1

0

1 I I I

2 3

rz= a

Fig (3·1) ILLUSTRAT10N OF A CENTRAL DIP IN

OPTLCAL FIBRES.

r

- 68 -

3.2 Theory

Refering to fig. (3.1), the refractive index distribution can be divided

into three regions.

In region 1, the refractive index variation is given by:

(3 0 1)

where

R1 r1

= A a

a is the fibre core radius

A is a fraction of the fibre radius (a), A« 1

n 0

is the refractive index at the edge of the core

0 is the relative refractive index difference between the

edge and the centre of the core, and denotes the dip height

a 1 is the index profile parameter in region 1.

In region 2, it is expressed as!

2 a 2 (1 - 2ll(R2

) 2j n (r) = n A a ~ R2 ~ a 0 (3.2)

where R2 r2 r2

= = a(l - A) a - aA

and a 2 is the index profile parameter in region 2.

In region 3,

where n2 is the cladding refractive index.

- 69 -

Consider the case of bound rays only, i.e. equation (2.19). Here as be fore,

two alternatives are to be considered.

1. b >- a (source radius larger than the fibre radius)

In this case the launching efficiency is given by:

n' = (3.3)

Since n(r) is expressed by·two different functions in the two regions of

fig (3.1), the above integral can be split into two parts and the launching

efficiency evaluated for the two separate regions as follows:

In region 2: the integral I2

is given by:

2 T r [n2(r) - n2

2 ] rdr I2 0 =

b2 A a Aa ~ r :=: a (3.4)

2 n 2[ a

But n (r) = 1 - 2ll( R2) 2] 0

= a(l - A)

(3.5)

= [ r2

26 a(l (3.5)

=

where

=

2 n

( 0 - n 2

2

n 0

2

- 70 -

) =

[ r; 2 _

Substituting and neglecting high order terms of A since it has been

assumed that A « 1

!2 4 T . 2 (~)2 [ i -( "2

1

A) "2 J = n 11 0 0 b

+ 2)( 1 -

Similarly in region 1, the integral r1 is given by:

2 T ra {.no 2 " Il

0 [ 1 - 2o < 1 - R

1) 1] = -

b2

where

4 T n 2o 0 0

!1 = b2

R 1

0

=

(ll

. 2

r 1

A a

"1 r 12 --+ • Aa

3

' "1 '· (ll -2

1) 3 l_ A2a2

4

n 2} r1dr

1 0

' "1 \ "l - l)(a1-

6

2)

T n 2o 4 [ r~ -

"1r1 cx1(a1 - l) rl "1 (al - l) (a 1 - 2) 0 0 = --+

b2 3Aa 8 A2a2 30

( 3. 7)

(3.8)

(3.9)

(3.10)

rl4 J I

. A3a3 dr

(3. U)

A a

=LJ A3a3 0

(3.12)

- 71 -

"'l - + 3

The launching efficiency is the sum of the two integrals r1

and r2

:

,. =

It can be seen that if o = 0 implying that A= 0 as well, the above

(3.13)

(3.14)

expression reduces to equation (2.24) which gives the launching efficiency

for b > a when no dip is present at the centre of the fibre.

2. b < a

This is the case where the source radius is less than the fibre core

radius and in this case the acceptance angle of the fibre is given by:

sin

and the launching efficiency is expressed as:

,. (3. 15)

note that the limits of integration are determined by the smaller of

a or b.

- 72 -

Evaluating equation (3.15) for the two regions as before; in region 2

the integral 12 is

= 2 To fb 1

2 2 b 0

=

=

=

n 211 r2 = 4 T 0 0

r { 1 -0

[ r 2 al2

a(l - A) }

[ r: - --'a;:._a_2_+_2 _____ a_21b

a (a2

+z)(l-A) o

b2 - _ _::.b _____ _

[

- 2 1].2 + 2

1].2 1].2 a (a

2 + 2)(1 - A)

[t l

(~) 1].2] a

(a2 + 2)(1 - A) 2

Similarly in region 1, the integral 11 is:

2 T

c A2 {n 2 1].1

n 2} 0 Il = ( 1 - 2 o (1 - R~ j -

b2 0 0

4 T n 2 A2a b al <afl) 2 L rl -= 0 0

b2 alRl + 2 Rl

where Rl rl

= A a

(3.16)

(3.17)

(3 .18)

(3.19)

r1dr

1 (3.20)

al( "I. - I)(al -2) 3 Jr dr

1

6 Rl 1 1

(3.21)

- 73 -

After some manipulation the above integral yields:

2 2 ll I = - 4 T n A o - -1 0 0 2

a ....!. (~) 3 a

al(al-l) b 2 + (-)

8 a

_ ~1 <a 1 -l)(a1 -2) 30

3 (~) ]

The launching efficiency for this case is therefore

= 1 a .

(a2 + 2)(1 - A) 2

al(al - l)(al - 2) b 3 J + -!..--...!-.....,....-....:"---(-a) }

30

It is also evident in this case that when the dip height o is zero the

above expression reduces to equation (2.25) which gives the launching

efficiency for b < a when no dip is present at the centre of the fibre.

Equations (3.14) and (3.23·) when plotted give Figs. (3.2 - 3.8) for

different values of dip height o and dip width A for a fibre with the

following parameters;

n = 1.47028

n2 = 1. 45 709

2A = 0.01794

a = 25 X lo- 6 meters

(3 0 22)

(3.23)

and a refractive index variation which is parabolic with a dip at the centre.

It is clear from the graphs that an increase in the dip height o or the dip

width A represents a reduction in the launching efficiency. For example

a dip width of 20% of the fibre radius, foro = ..r._ and 4 '1. = <Yz = 1, represents

- 74 -

a 5% reduction while for o = ~ the reduction increase to 8%. This

reduction in the launching efficiency represents an increase in the

coupling loss as shown in Fig (3.9) .

•

n' r-< " c: "

. '. ~-' (_' .

() --, t..

::r' /_ ~-·

:l

"" "' "' "' ,... n '• ,... (1)

" n '<

' ,, <

'·

Fig. (3.2) Launching Efficiency V.(~) a

! "".o; ' ., .. -) ~

0 (',

= 4

1 - A = l%a 2 A = 10%a fpr "1 "2 = 1 ...,

2)%a V>

3 - A =

.l

n'

t-<

"

. t: •

.. ·{ . ..

c: ., . -~ :1 .. 0: •• n :s

'.-. ~- . '

···,

Fig. (3.3) Launiching b

V. (-a) Efficiency

.. ,

0 =

1 - A

2 - A

3 - A

!:,

2

=

=

=

1%a

10%a

20%a

for "1 == f'/2 1

- ·--., ';•"""';>

.._ '-· . ( ' . ~.)

. ('• '· .. ·.,

.. '>

n'

-·, 1 t..' .:

r 2 " c: :l 3 n ;:J"

:l ' "" "' H> H> ~· n .... ro :l n '<

.... _,..-- ......... .

' ':) _, ')

Fig. (3.4) Launching Efficiency V (~) . a

0 = 11

- A = 1%a

A = l0%a

- A = 20%a

for "z = "1 = l

-···j·--.... -- ,.,_, ______ . ___ . _______ , __ - ...... ___ -----..

' r· j 7)

. •:. ~·

b x·:r.·-· -+(-)'···:J

a

·, ' '-·

..... .....

. ,_.

n

·•-t:• .... ~)

·, ' ~· '·· -

·cc ,_.

Fig. (3.5)

. ------------5, ' t 8

b h . g Efficiency V.(_) Launc ln a

··--.-·-1~

'~

/; = 4

1 - A = 1%a

A = l0%a 2 -

A = 20%a 3 -

1 : is

n'

r " c ~

" ,. ~· ~

"" "' ,.., ,.., ~·

" "· " ~ ()

'<

.'..!" . <. -,

--·. ,-, ._,-(_: .:

.... c.: ' . ..)

··. r1 . -- (_ .

<. ·-·

~::.

.-;. (,

.. •

1 - A

2 - A

3 - A

.. .

=

=

=

=

to 2

l%a

l0%a

20%a

. .................. ·----·-- -·-------·---------·-..,--·----------- -------- .,..-----------·- ... -----,---·-- ....... ----------------------- ----------·· ·- ... ·-- ·-- -- ·-- --:-·-··--------· ------·-:-·---------·- ----·-, ·'~- 5 8 i 3 1 2 i ! 1 S ; 8 ~: 0

Fig. (3.6) Launching Efficiency V.(~) a

X~,._.- I

I .. ;

n'

:< ~ ~~· -· 3 ·1 :? .,

t-·

··: t. (l ._) ,.•

··-c.· . ._)

•' < ..

•:· .-:.·

-~ ...;......,

-~---~-~ tl -.... ,_ ....

0 --~

"· . . ·-, --~-

.,

Fig.(3.7) Launching Efficiency V.(~) a

'r. I,;. •\ •')

1

2

3

.s - A

- A

- A

.!

= b.

= 1%a

= 10%a

= 20%a

1 J

for cx1 = cx 2 = 2 "' 0

-~ .. ·' -...' (. • t

c ; -

n

-.. :· ' • ..

r " c ~ n ;:r ,... ::l

'"' "' "' ..,., ,... n

"' ~ :. n ·- -· "'

'-.

Fig. (3, 8)

= (:,

4

a,b,c, for o. 1 = d,e,f, for "l =

·-~--·--·----;---··- -·-·--------········--· ""' "' .. ----··-··--------·--·-·- ----·---·- .. - ........ .

s 8 !J

Comparison between launching efficiencies V.(~) . a

1S

"z = 2

"z = 1

·-·-·-----------·· ··----·----,

b ->(-) a

: 8 .:~c;

X ~ c~-1 ' ,,

25

25

24

23

22 r 0

"' "' ~· 21 ::l

~

0..

"' 20 ~

19

18

17

15

15

1 4 0 2

Fig. (3.9)

10 12 Coupling loss V.(~) for various values of o and A a

0

1 4 1 5 18 20 xur1

00 N

- 83 -

3. 3 Discussion

The fact that pulse spread can be reduced dramatically by quadratic grading

of the refractive index profile led to the expectation that if the refractive

index profiles were optimised, a further reduction in intermodal pulses

spread might be achieved. Thus the investigation of fibres having arbitrary

shaped refractive index profiles has been prompted by this expectation.

The manufacturing process of optical fibres involves the presence of two

dips (valleys) in the refractive index profile, one is at the core-cladding

boundary and the other at the centre of the fibre core. The former is

made deli·berately during the manufacturing process and serves two

purposes:

1. Reduction of internal mechanical stresses due to the gradient of

the dopant concentration,

2. The difference between group velocities of modes is reduced to a

minimum, hence pulse dispersion is reduced.

The latter is unavoidable and its presence is due to the high temperatures

involved during the manufacturing process.

Its effect on the launching efficiency of optical fibres has been investi-

gated and analytical expressions have been obtained. It has been suggested

b . . ' ( l3) h h f d" . h f . . d f"l y P. D1 V1ta t at t e presence o a 1p 1n t e re ract1ve 1n ex pro 1 e

in the centre of the core does not produce appreciable differences in the

launching efficiency. However, the present analysis shows that it does,

and a reduction in the launching efficiency of 8% has been demonstrated.

It has also been found that, for a certain dip height and dip width,

- 84 -

increasing the refractive index profile a1 improves the launching

efficiency. Therefore it would appear that a minimisation of the

central dip effect on the launching efficiency of an optical fibre may

be obtained by making the refractive index variation in the dip region

an a-power law profile.

- 85 -

CHAPTER 4

EFFECT OF CENTRAL DIP IN THE REFRACTIVE INDEX PROFILE ON BIREFRINGENCE

IN GRADED INDEX SINGLE MODE FIBRES WITH ELLIPTICAL CROSS SECTION

4.1 Introduction

Single mode optical fibres offer greater transmission bandwidth than

11 h "d" d" (l) a ot er gu1 1ng me 1a . Their bandwidth is limited by material

dispersion and waveguide structure. These limitations may be reduced

or eliminated by choosing an optimum wavelength such that the material

d2n 2 dispersion either vanishes ( /dA = 0), or compensates for mode

do o (2,3) lSperSlOO .

If circularly polarized light is launched into such a fibre, the state

of polarization of the light changes along the fibre length. This

is'due to the different propagation velocities of the polarization

components of the light with respect to the axes of the ellipse, i.e.

h f "b b" f 0 0 (1) t e 1 re acts as a 1re r1ngent med1um . This birefringence is

the result of a difference in the values of 8 for the propagation

constants of the fundamental mode. This difference in the values of 8

causes the fibre to exhibit a linear retardation, the value of which

depends upnn the fibre length. The resulting polarization state varies

periodi-cally along the fibre with a characteristic length "L" where

L = delta 8

Typical single mode fibres were found to have a period of a few centi­

(4) meters .

- 86 -

The effect of different central dip heights (o) in the refractive

index profile of graded index single mode fibres with elliptical cross-

section is investigated.

4.2 Theoretical background

Several calculations of the birefringence due to core ellipticity

have been performed using vector (5) and scalar (6) perturbation

methods.

For weakly guiding (6 << 1), slightly elliptical (e2 << 1), step index

fibres, the difference in phase velocity (delta 8) between the two

polarization states of the fundamental mode is given by (6);

deltaS =

where

I~ =

V =

v2 =

a k (n2 0 0

-(u2 + w2)

2)~ n2

2 J (U) 3] + U W { ~ (U)}

1

a is the fibre radius, J 0

and J 1 are Bessel functions of the 1st and

2nd kind respectively.

2n = A is the wave number in free space

n0

& n2 are defined previously

(4. 1)

(4.2)

(4.3)

(4.4)

(4.5)

- 87 -

A monomode graded index elliptical fibre whose major and minor axes are

a and b respectively has a core refractive index of the form

2 n (r,e) = n 2 [1 - 2llf(r,e)J

0

where

and a cladding refractive index of the form

r and R is the normalised radius a

(4.6)

(4. 7)

For fibres having such refractive index profiles, the phase difference

(delta S) is given by (7)

deltaS = lji' J] 0 cladding

+ fl R f'[ lJ! (lji' + ~ 1jJ - l lji') 0

o2R240

(4.8)

where the prime indicates differentiation with respect to R and

No r 0

=

lji0 is the scalar field in circular fibres and is found by solving the scalar

field equation (8).

f(l) indicates that the evaluation of the fields is being carried out

in the cladding side of the boundary.

For step index fibres f(r,e) = 0, and equation (4.8) reduces to:

- 88 -

deltaa 1/!'1/! +! (l/1'2 + 1/J" - 1/!o'>] 0 2 4 0 0 (4.9)

with l/10 and w2 given by (6): i.e.

J (UR} 0

l/10 = ...:::.J-;----0

2 e

- 2T 0

u2 RJ (UR) --

1 4 28 ] (4.10)

K (WR) 0

2 e

1/J 2 - --"'K:::---- 2K 0

[ dK0

(WR) w +-2 cos

0

where R is the normalized radius ( !:.) a

a is the azimuthal angle

U and W are the arguments of the Bessel arid modified Hankel

d and c are normalization parameters and are given by (9):

- 1

and

c - d =

If the function fin equation (4.7) is expressed as

f = "2 R

functions.

where the first term represents the normal a-profile refractive index

and the second term represents the dip at the centre of the fibre.

Then, following the same analysis given in (7), 1/J is obtained in the 0

(4.11)

(4.12)

(4.13)

( 4. 14)

same manner as before. <)> 2 in the cladding is given by (4 .11) and in the

- 89 -

qladding is given by (4.11) and in the core w2 is the solution of the

equation

w" + .!. w' 2 R 2 2 4 2 + (U - - - V f)

R2 = 0 (4.15)

A series of numerical calculations were performed to calculate w0

and w2

and these values were substituted in equation (4.8) to calculate delta

for different values of dip height.

1 2.

1 1

1 0.

9

8

7

~-~ 0.. NI!> 6. C>~ ~ ... Wll> ---.

'"" NU>

5 0

4

3

2.

0+--+----,------.-------.-------.------,-------.-------,------.-------.------~ 0 2 3 4 5 6 7 8 9 10

_,. V Fig. (4.1) Normalized phase difference deltaS as a function of V

1 2

1 1

1 0.

9

8

7 " a.

(1) (1) ...- 6 . ~ " 1>0> ~

w"' "' ......

5 ~ N

4

3

2.

0+--+--~------~-----.------.------.-------.------.------.------,------, 0 2 3 4 5 6 7 8 9 10

Fig. (4.2) Normalized phase difference delta S as a function of V _,. V

1 2.

10

9

8

" 7-"' '" 0.. ~"' N >-'

t> ,-, 5 ~" . w

--- "' N

5

4

3_

2.

L

I

n I

0 0!--~---,------2,------3,------4.------.5------,6------~7------~8------~9~----:,0

Fig. (4.3) Normalized phase difference delta S as a function of V calculated from equations (4.1) and (4.9).

.,. V

-~ i. ' I

10 I 9.

00

8 8

6

7 4

"' " "2 = 2

al = 2 N 5_ ~ 0.

"'"' <>::: 0 = A

2 ~ " "' 5 ...... "' N

"' w

4

3.

2.

1.

0_ 0 2 3 4 5 6 7 8

.,. V Fig. (4.4) Normalized phase difference delta S as a function of V for various dip index profile (al)

11.

10

9

8

7.

6 (1)

" N ~ 0. '" (1) 5 -1:3

.... ,., w " -- m

4

3

2_

1 .

0, 0 2 3 4 5 6 7

Fig. (4.5) Normalized phase differecce delta 8 as a function of V for various dip profile parameter ( a1)

8 9 -+V

- 95 -

4.3 Discussion

The origins of birefringence are twofold.

l. Ellipticity in the fibre core establishes preferred fast and slow

axes of propagation and produces a retardance which depends on the

d f 11. . . (3)

egree o e 1pt1c1ty .

2. The presence of an asymmetrical residual stress distribution within

h f . b 1 . b. f . ( 10) t e 1 re, resu ts 1n stress 1re r1ngence .

A number of investigations into the effects of core ellipticity on the

operation of single mode fibres have been carried out both in tele-

communication systems(ll) and in devices such as Faraday-effect fibre

(7 8) current transducers ' .

Fig. (4.1) show the variation of the phase difference (delta S ) as a

function of (V) for a step index fibre (Equation 4.9). This was corn-

pared with values obtained for (deltaS) using equation (4.1) There is

a slight difference between the results obtained. This may be due either

to approximation error during the computation process, or to the approxi-

mation involved in the derivation of the scalar modes by solving the scalar

wave equation in elliptical co-ordinates and then correcting the scalar

propagation constants to include the effect of the difference between a

circle and an ellipse in the refractive index term. In this extra region

the fields obtained by considering a cllrcular system give an ~nsufficient

approximation for the case of an elliptical system, because the radial corn-

ponents of the two fields are discontinuous along the two different boundaries.

It can be seen from the graph that the circularity required to produce lo~-1

birefringence for current-transducer application is highest near the

- 96 -

preferred operating point for a single mode fibre. Lowering V or

increasing its value towards the multimode region relaxes the tolerance

on circularity. The expression obtained for calculating the phase

difference suggests that larger values of ellipticity can be tolerated

in fibres having small (6), but again the index difference (6) cannot

be reduced too much because radiation loss due to fibre bending becomes

intolerable when (6) is excessively small.

The presence of a dip at the centre shows that the higher the dip height

(o) the lower the maximum value of the birefringence. Making the

refractive index profile in the dip region an a profile does not appear

to have much effect on the birefringence compared with the effect on the

launching efficiency as explained in the previous chapter, i.e. the

shape of the dip profile is not important but the depth o of the dip

is the governing factor.

- 97 -

CHAPTER 5

EFFECT OF THE CENTRAL DIP IN THE REFRACTIVE INDEX PROFILE

ON PULSE BROADENING OF MULTIMODE OPTICAL FIBRES

5.1 Introduction

Use of optimum refractive index p·rofile can reduce the intermodal

delay differences in multimode optical fibres (1, 2) so that pure

material dispersion becomes the dominant source of pulse broadening.

Thus multimode fibres with optimum or near optimum refractive index

profiles represent an attractive transmission medium for high bandwidth

optical communication systems.

Modal delay distortions occur because the many different modes travel

at different group velocities. This results in spreading an impulse

response over a time interval that is equal to the difference between

the arrival times of the slowest and fastest modes. This pulse

d . . . db d . . . 1 b d "d h ( 3) sprea 1ng 1s accompan1e y a re uct1on 1n s1gna an Wl t s .

Since it is difficult to make index profiles of optimum shape, it is

important to know how any departure from the optimum affects the impulse

response of an optical fibre.

One of the problems encountered in fibres produced by the Modified

Chemical Vapour Decomposition (MCVD) method is the drop-off of the

index value (step distortion) near the core cladding interface as shown

in Fig (5.1). This represents a deviation from the optimum index

profile.

- 98 -

A second ~~Lation that may be encountered in fibres produced by the

same method is the depression in the index distribution on the fibre

axis as shown in Fig. (5.2). This is due to the evaporation of the

dopant material from the innermost layers during the collapsing process

and is caused by the high temperatures involved in the manufacturing

process.

Calculations of impulse response and profile dispersion of optical

fibres have been carried out by many authors. R H d N. 1 . (4) . anson an 1co a1sen

tried to predict the impulse response by fitting the refractive index

profile by an a-profile, but have indicated that this gives rather

inaccurate results. A more general approach has been used by Hansen (5)

and Marcuse ( 6). The group delay of each mode is calculated using the

WKB(l) approach, and the impulse response is obtained by counting the

number of modes arriving at the receiving end of the fibre in a given

interval of time. These methods involve the calculation of two

caustics for each propagating mode. Chu( 7) has proposed a method of

calculating the impulse response, without the need of calculating

the caustics. In the following, a review of the theory proposed in

Reference (7) is given and its application to arbitrary refractive

index profiles with deviations from the optimum >s investigated. The

effect of these deviations in the impulse response of the optical

fibre is demonstrated.

- 99 -

Fig (5.1) ILLUSTRATION OF A STEP DISTORTION IN

THE INDEX PROFILE OF OPTICAL FIBRES

- lOO -

n (r l -...,---no·--_

Y\1

6

0

--I I I I I I I I

------1---- ----~----

r

Fig (5.2) ILLUSTRATION OF CENTRAL DIP IN THE REFRACTIVE

INDEX PROFILE OF OPTICAL DIBRES

- 101 -

5.2 Theoretical review:

The refractive index profile of an optical fibre is given by:

(5.1)

where F( A,r) is any function of radius r and wavelength A and n0

is

the refractive index on the fibre axis.

The propagation constant a for the mode (~,v) can be expressed as

= (5.2)

where P is the mode parameter which varies between zero for the

lowest order mode and 26 for the modes whose phase velocities coincide

. h h f l . h l dd' (S) w1t t at o a p ane wave 1n t e c a 1ng .

k = is the free space propagation constant.

The wave propagation in an optical fibre is given by(9 ): ·

(2~ + V + 1) .!f. = 2 J

rl

0

/p - F (A,r) dr

where r 1 is the caustic that makes the radical zero, i.e. is the

solution of the equation

P - F 0; r) = 0

(5.3)

( 5. 4)

- 102 -

Let the.principal mode number be m then,

m = (2!1 +V + 1)

and equation (5.3) becomes

m = 2 1T

N k 0

lP - F(>,,r) dr

(5.5)

(5.6)

In reference (7), it has been shown that equation (5.6) can be evaluated

without the necessity of computing the caustic point r1

. This was

achieved by obtaining G(>,,F) which is the inverse of the profile function

F(>.,r).

Obtaining the inverse profile function G(>.,F) and changing the variables,

equation (5.6) becomes

m = 4 1T

n k 0 fo

p

,lP - F G' (F) dF (5. 7)

where G'(F) is the derivative of the inverse profile function with

respect to F.

The above integral can be expressed as the convolution of two functions;

lP and G'(P), where G'(P) is the derivative of the profile function.

4 m =

1T n k ,lP ,, G'(P)

0

where* denotes the· convolution operation.

(5.8)

- 103 -

The normalized group delay T for the modes of principal mode number

m can be obtained by differentiating equation (5.6) with respect to k

and substituting for m from equation (5.8). This gives

1

1 ~(P)* ,lp T = - 1

.; ( l -1

P) G'(P)* lP (5.9)

where:

~(P) [1 + n1

-D2

- p(l + Dl)) G' (P) = 2 (5.10)

n 0

A d n A dfl Dl

0 = ---n dA 2A dA 0

(5.11)

= (5.12)

n1 and n2 are the dispersion parameters.

The impulse response f(T) is given by( 7);

f(T) 4 dm2 dP = M dP - dT (5.13)

where M is the largest principal mode number given by(lo>;

M r Cl. a 2k2n2fl r =

Cl. + 2 0 (5.14)

- 104 -

For fibres having refractive index profiles of the form shown in

Fig (5.1) which are composed of a power law and a linear function

i.e.

F( >., r)

and

(r - ro) -FCA,r) 26 =

(r 0

0 ~ r ~ r 0

r (a - r)(...£)"

a

- a)

(5.15)

r ~ r ~ a 0

The impulse response has been reported previously( 3 •7? It has been

shown that the step distortion produces a pulse tail of low amplitude,

and if the step extends deeper into the core, the normalized group

delay T becomes larger, the tail becomes longer and the main part of

the impulse, which is determined by the power law region, becomes narrower.

This justifies the presence of the valley in the index profile near the

core cladding interface mentioned in Chapter 3, which serves to

minimize this effect.

Profiles of the form shown in Fig (5.2) are composed of a linear

function in the depression region and a power law as follows

F(>.,r) = 26 (.!:.) a

et

Aa ~ r ~ a

2 2 2

F(>.,r) l n3 nl - n3

(E_) 0 ~ r ~ aA = -2 2 a A n n

(50 16)

0 0

where A is as defined previously.

- 105 -

In the above definition the depth height o is determined by the

difference between n0

and n3 while the one considered in Chapter 3

was the difference between n1

and n3' The reason for that is that

in the present analysis the aim is to extend the power law profile

to the fibre axis and not to reach a step index as was mentioned in

the previous chapter.

For such a profile given by equation (5.16), the principal mode

number m and the normalized group delay

region as follows:

(l) For the power law region

G(>.. F)

and

= 1

(!...) /a. 28

G'(>.,F) l .!:. - l

= ..l. (.!.._) I a. F " a. 28 .

and (2) for the linear region.

F(A, r) = l -

2 n3 -

2 n

0

= l - - F

(!..._) a A

are calculated for each

F ~ 28 (5.17)

(5.18)

0 ~ r ~ aA (5.19)

(50 20)

- 106 -

2 2

( ~~ 2 nl - n n3 3

1 - - F = 2 2 n n 0 0

[1 -2

- F ] 2

n;) (.!:) n2 A n3 (nl = a 0 2 n

0

2 2 (.!:)

no A [ 1 -

n3 - F J = a 2 2 2 n - n3 n 1 0

the inverse profile function for the linear depression in

the index profile at the fibre axis is given by

n2 A 2 0

[ 1 -n3

] G(A,F) = - F 2o ~ F ~ 26 2 2 2

nl - n3 n 0

where 2 2 n - n3

li = 0

2 2 n 0

Substituting equation (5.17) into equation (5.8) gives for the

principal mode number m

4 m = n k

0

2 + <X

P 2a

for the power law profile region.

B(l + 1 <X

.!.){1-I(l 2 X

1 +­a'

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)

- 107 -

Substituting equation (5.24) into (5.8) gives:

4 m = - n k 1T 0

for the depression region. where

2 + Cl

P--:za B (l + i .!.){ l - I (l a' 2 x

l +­a'

( 1 - X) 2 3t ]

B (l l +­Cl

l ) 2

is the Beta function

X = 2t, {A)a p

The normalized group delay for the two regions is given by( 7)

and

l t = --=---

for

T =

for

,f""l:-p

--=-1-{(1 ,tl7P"

1 l-I(1+a

- ( 1 + 0 1) ~ :. 2 p X l l ~ 1 - I (- -)

X a , 2

where I (.!. -21

) 1s the incomplete Beta function. X Cl

2M ~ P ·~ 2

<5.26) I

(5.27)

.!.) 2

} - l

(5.28)

(5.29)

- 108 -

If the dispersion parametemD1 and o2

are assumed to be zero then

equations (5.28) and (5.29) becomes:

T =

and

T =

1

~p

for 2li{A)"' ~ P ~ 26

1

for 26(A)"' ~ P ~ 26

1 +­OL'

.!.) 2

i>] ]- 1 (5.30)

(5.31)

The impulse response has been calculated using equation (5.13) for

a fibre with the following parameters:

n = 1.47028 0

n2 = 1.45709

6 = 0.00893

Cl = 2

a = 25 11 m

-6

10

9

8

A = 10% a 7

0 t. = 4

Q) 6 " " 0 c..

" Q) .. 5 Q)

" .... ::J

~ 4 H

3

2

1

-5 -4 -3 -2 -1 0 Normalized group delay

Fig. (5.3) Impulse response of power law profile optical fibre with central dip.

.... 0

""

2 3 T X10-9 sec

---------=--=-:_:_:--::_::_-________________________________ _

7

10.

9.

8.

7

., "' 6 " 0

"" "' A= 10% .,

a ... ., 5 . !J. "' 0

,... =- " 2 "" e ....

4

3.

2.

1

~~~~---T---·--~~~<Iv ' ~6 ~5 -4 -3 ~2 -1 0

Normalized group delay Fig. (5 .4) Impulse response of power law profile optical fibre with central dip

~

'"' 0

9

8

7 A = 10% a

0 = !'J. "' 6 " " 0

"' " "' ... 5

"' " ,... ;:J

"' .... .. .... H 4 ....

3

2

-8 -7 -6 -5 -4 -3 -2 -1 0 2 3 Normalized group delay + T X10-9 sec

Fig. (5.5) Impulse response of power law profile optical fibre with central dip.

-

9_ ((( l\\ i ~ I

8 '

A = 10% a 0 1!.

I =-4

7 I I 0

1!. I =-

I 2

<l) 6. ( 1-o = 1!. "' " I 0

"' \

"' <l) ... 5 <l) it "' I ....

' " I "' -s ' -H 4 ' N

'

3 ' I I

I I

2 I

1

~?QJ ~~ ~ ~ ---8 -7 ..:.s -5 -4 ..:.3 -2 ..:.1 0 i 2 3

Normalized group delay -> T X10-9sec

Fig. (5.6) Variation of the impulse response with central· dip height (o) .

. - ~---- - .

\

9

8 A = 15% a

0 = ~ 7 4

QJ ., c

6 0

"' ., QJ ... QJ ., 5_ ,....

" ~ H

~

4_ ..... w

3

2_

lJ 1_

--~ ' c::l I

_:_3 _:_2 i -7 -6 -5 -4 -1 0 sec

Fig. (5.7) Impulse response of power law profile optical fibre with central dip.

-6 ..:.5

A = 15% a

0 = ~ 2

_L

-3

~

-2 ..:., Fig. (5.8) Impulse response of power law optical fibre with central dip.

9

8.

7.

6

5

4.

3

2

1. V c:l

0 i :2 3 + X10-9 sec

r \ 9

(

A = 15% 8

cl t. =-4

7 -cl t. =-., 2

"' " 0 s_ "" "' ., ... ., "' .... 5_ " "" = I H

' ' 4 ... ...

V>

' 3_

2.

w ~ ~

~ ~

,. Vl

I • ~6 .:.4 T 2 3 -8 -7 -5 -3 -2 -1 0 1

T X10-9 sec

Fig. (5.9) Variation of the impulse response with central dip height cl)

6

5

4

3_

2_

-5 -4 -3 -2 -1 0

Fig. (5.10) Variation of impulse response with central dip width (A)

9 (/1' 0

!::. = 2 p 8. r

A = 10%

7. A = 15%

6.

5.

I

4 .... .... .... I

3

2.

·~ ~V -= ~ rv ..,...__

~ ~ ~ .. -8 .:...7 -6 -5 ' ' -3 -2 .:...1 0 1 2 3 -4

T X10-9 sec Fig. (5.11) Variation of impulse response with central dip width (A)

- 118 -

5.3 Discussion

Assuming equal excitation of all modes and no mode coupling, it

can be seen from the graphs that the impulse response depends

primarily on the shape of the refractive index distribution in

the core. A dip in the refractive index profile, characterized

by its depth o and its width A, affects the impulse response,and

the response to the left of the origin in the graphs is due to

the presence of the central dip. Thus the presence of the linear dip

increases the pulse dispersion and this represents a reduction in the

bandwidth.

It can be seen from the graphs that both the depth and the width of the

central dip increase pulse dispersion. Wider depths have drastic

effects on the impulse response and hence produce greater pulse

dis pe rs ion.

- 119 -

CHAPTER 6

EXPERIMENTS

In this chapter some experimental results related to the theoretical

content of the thesis will be reported.

Experimenting with fibres is expensive due to the high precision

devices required for alignment etc. Additionally, very good resolution

is needed to detect the effects of parameters such as non-uniformities

in core diameter or the refractive index profile.

The equipment available for this project was rather limited. Hence

not all the mathematical derivations could be tested experimentally.

Also, the industrial manufacturers of optical fibres were reluctant to

provide values for the parameters of their fibres. Specifically no

exact values were given of the refractive index exponent a, the width

and height of the central dip or the ellipticity of the fibre.

The suggestion was that the fibres were circular and had a refractive

index variation which was parabolic and the attenuation was 2.5 dB/Km,

hence the assumption that a = 2 throughout this work.

- 120 -

6.1 Fibre end preparation:

One of the most important criteria when dealing with fibres is

the ability to obtain terminations of acceptable quality. Optical

fibres have to be externally excited. Therefore, for maximum

light to enter the fibre the scattering by the fibre end must be a

minimum which means that the fibre end should be as optically flat

as possible.

Throughout the recent progress in the field of fibre optics, two

types of fibre to fibre connection have come into existence. Fusion

arc splicing which may be essentially regarded as the welding of

fibres with index matching liquid in between. This is considered as

permanent mounting and provides low splicing loss, typically 0,5 dB(l),

Before welding is carried out a clean and flat end is essential for

a good seal.

The other type that provides more flexibility is to have the fibre

end free and accomplish the coupling via precision alighment. Convent-

ionally 'V' grooves are employed for the task. Whether the fibre is -

protected by surrounding layers or lies bare in the groove would

depend upon the pa•ticular requirement.

Much progress has been made in breaking fibres to produce reasonably

flat terminations. Gloge(2

) has indicated that to avoid the occurrence

of what is called a beak, a known stress must be applied to keep it under

- 121 -

tension while simultaneously bending it over a curved surface.

Another type which was used earlier consisted of curve shaped. foam

over which the fibre was stretched to the correct tension by a dial

gauge. A knife fixed on to a pole would then come down and initiate

a crack on the fibre. Another approach is to tape the end of the

fibre over any clean curved surface and pull it by hand. Touching

it slightly with a knife edge would then effect the cleavage.

The latter method was used here and steps given in reference (3) were

followed so that the fibre end was ready for polishing.

This was accomplished as follows:

Two discs were mounted on opposite ends of a small D.C. motor shaft

whose speed could be varied using a variac. On one disc the rough

polishing pad was fixed and on the other the fine polishing pad. The

fibre and the protecting sleeve around it was pushed through a small

hole on a mount facing the rotating disc and at right angles to its

plane. This arrangement ensured that the fibre was being polished

in a plane perpendicular to the fibre axis, since a sloping end is

undesirable as it reduces the gathering power or collecting efficiency

of the fibre ( 4 ).

The rough pad was used to reduce the protruding section and at the

same time the process was observed-under the microscope. When this

was completed, transferring to the fine polishing pad finally yielded

the smooth surface desired. The motor provided a faster and easier

method for polishing than polishing by hand. The arrangement is shown

in Fig. 6.1.

I N N ..... I

Fig (6.1) Fibre and preparation arrangement

' M N .-<

'

Fig, (6.2) General view of experimental assembly

- 124 -

6.2 Measurements

The equipment supplied for this project was as follows: (see Fig.6.2)

l. Optometer (analogue) plus a separate opto-detector from

United Detector Technology.

2. Micro manipulators, three rotational and three translational

stages with resolutions of 5 ~m from Unimatic Engineering Ltd.

3. Light sources:

a - He-Ne laser,

b -Two LEDs,

1.= 0.632 nm

1.1

= 660 nm, = 890 nm 2

4. Ealing Beck microscope and Kyowa viewing microscope.

5. Various convex and concave lenses and beam splitter

6. Optical bench and cast iron bed on an anti vibration mounting.

The hardware constructed was as follows:

l.

2.

Mounting arrangement to fit onto the micro manipulators. These

comprised two plates, the lower one was to join the assembly to

the· micropositioners. The other plate was mounted on the top.

Two interchangable blades were also provided to be used on the

upper part, one to hold the ferruled terminators and the other

for bare fibre ends. 'V' grooves on these plates were cut

accordingly. A clamp device with rubber attachment on the lower

side was used to hold the fibre down.

A housing box and a drive unit for the LEDs. The circuit

operated from a ~741 operational amplifier and two intensity

control buttons were provided.

- 125 -

3. A differential amplifier with variable gain. This was used to

provide a digital read-out from the output of the photodetector.

This also used a ~741 amplifier.

4. Various mounts for examining fibres under the microscope and to

couple the fibre into the optometer head. These were cylindrical

structures with holes bored through the middle. The mounting

device for bare fibres was in two halves each one being individually

'V' grooved. The various items were aligned on an optical bench.

Measurements, using the He-Ne laser as a light source on the variation

of the launching efficiency with axial displacement and transverse

offsets were explained in (3). Therefore the measurements carried out

here were devoted to LED as a light source (incoherent source), and

the variation of the launching efficiency with separation, lateral

displacement and misalignment was measured. This was carried out

using three different lengths of the same fibre. The lengths used

were 1 meter, 9 meters and 540 meters, Variation of the launching

efficiency with the variation of the light source area using a variable

slit could not be verified for the obvious reason that reducing the

1. h . d . d d b h d. ff . 1' . ( 5) d d . 1g t source area 1s ec1 e y t e 1 ract1on 1m1t an re uc1ng

it further will give inaccurate results. Microscopic objectives were

used to collect and focus the light onto fibre-;- The emitting surface

area of the LED was made equal to that of the fibre core diameter.

Index matching liquid was used to minimize reflection losses.

2 0 -r----.._

1 9.

<=

"' 1 8 u

" "' .... u ....

1 7. "" "" "' eo

" .... 1 6 .c

u ~

" N

" "' "' ..l

1 5

1 4

13

1 2

11~------.-----.------.------.-----,------.------.-----,------.------.-----.------0 2 3 4 5 6 7 8 9 1 0 11 1 2 ->(!) a

Fig. (6. 3) Measured variation of the launching efficiency with separation for different optical fibre lengths

b 1 - =

1 8 a

.! = 1 a

16 a = 2

>, 1 4 R. = 1 m u c Q)

R. 10 m .... = u

1 2 . .... ..... ..... R. = 540 m

"' "" c 10 ....

-5 .... N c "' " " ,..,

8

6

4

2

0.1!------.-----.-----.-----.-----.------.-----.-----.----~==~~ 0 2 4 6 8 10 12 1 4 16 20

Fig. (6.5) Measured variation of the launching efficiency with displacement

-----~============~~======~--------------------

10

8_

6

4

2

04]~~~~~~~~-~ 0 2 4 6 8 1 0 1 2 1 4 1 6 1 8 20

Fig (6.6)

--------

Comparison between the variation of calculated and measured launching efficiencies with displacement

~ X10-1 a

' 20

1 8

1 6.

"' u c:

1 4

.~ 1 2 u ·~ .... .... "' o/)10 c: ·~ ..c u c: ~ 8 ,..,

4

2

0, 0 2 3

Fig (6.7) Measured variation of the

b -- == a

s - = a

Cl =

4 5 6

launching efficiency with misalignment

l

l

2

7 8 _,. a 0 X101

9

.... w 0

20.

1 8

16

"'14 u c

"' ·.< u ... 1 2 "' "' w

6

4

2

measured(~ = 1 m)

measured(~ = 10 m)

~.....,_-n measured(~ = 540 m)

b a

s a

a

= 1

= 1

= 2

041----------.---------.---------.---------~--------.-~------,----------.---------r--------~ 0 2 3 4 5 6 7 8 9

Fig. (6.8) Comparison between the variation of the calculated and measured launching efficiencies with misa1ighment

.,. cro XHJ1

- 132 -

6.3 Discussion

Throughout the measurements, care was taken to keep the fibre ends

clean and free from dust. 'Inhibisol' solvent and 'Omit plus'

sprays were used for this purpose.

Due to diffraction limitations, verification of the variation of the

launching efficiency with the variation of the light source surface

area, was not attempted.

The slight discrepancy between the theoretical and practical results

could either be due to the exact value of the refractive index

profile exponent a not being equal to 2, as stated by the manufacturer,

or to the presence of the central dip in the refractive index profile

which was not taken into account in the theoretical analysis.

- 133 -

CHAPTER 7

CONCLUSIONS

The major object of this report is to study the effects of various·

fibre parameters on the source-fibre coupling using the familiar

approach whose theoretical basis has been established, namely

geometrical ray theory.

Coupling theory, which plays an important role in the study of

optical fibres, was presented in Chapter 2. The investigation

covered guided rays only and the contribution of leaky rays was

taken into consideration later.

Simple analytical expressions were obtained for both a-power law

profiles and inverse a-power low profiles.

It can be seen that for LED-Fibre coupling, when the source area

is larger than the fibre core area, direct coupling represents the

best coupling condition, and the use of self focussing lenses or

micro lenses suggested by many authors appears to have no effect on

improving the coupling efficiency. The use-of such devices may

improve the angular properties of the emitted light so as to decrease

the broadening of the temporal pulse along the fibre. When the core

area is larger then the source area, the coupling efficiency can be

improved up to its maximum value when a lens is placed between the

course and the fibre. This improvement will result from co.llimation

- 134 -

as the source is magnified to fill the fibre core cross-section.

It was found that the launching efficiencies for both a-power law

profiles and inverse a-power law profiles, approached that of the

step index fibre as a limiting case. Unlike a-power law profile

fibres, inverse a-profile fibres offer minimum launching efficiency

at the.core centre but this increases gradually towards the core

cladding interface. This is due to their acceptance angle and their

numerical aperature which is a function of the dip height at the core

centre.

In multimode optical fibres, the presence of small ellipticity in

the fibre core, was found to improve the launching efficiency of

parabolic index (a= 2) fibres.

This improvement is significant only in parabolic index fibres, and

in these fibres an optimum value of ellipticity can be found that

improves the launching efficiency without affecting the fibre bandwidth.

This is of practical importance since the values of a in the region

a = 2 are the values of main interest as the optimum profile for

minimum pulse dispersion is close to parabolic.

The high temperatul'e collapse of fibres produced by the (CVD) method

yields a refractive index dip on the fibre axis. The\effect of

this dip on the launching efficiency of power law optical fibres has

been investigated and analytical expressions, relating the launching

efficiency to the dip height have been obtained. It was found that

I

- 135 -

the dip height affects the launching efficiency adversely and this

effect can be minimized by making the refractive index variation

in the dip region an a-power law profile.

Regarding birefringence in graded index, single mode, elliptical

fibres, the expression obtained for calculating the phase difference

shows that the higher the dip height (o) the lower the maximum value

of the birefringence. Making the refractive index profile in the

dip region an a-profile does not appear to have much effect on

birefringence compared with its effect on the launching efficiency as

explained in Chapter 3, i.e., the shape of the dip profile is not

important but the depth o of the dip is the governing factor.

Increasing the dip height increases the position of maximum birefringence

and reduces its magnitude,

Previous investigations reported in the literature have shown that

any deviation of the refractive index of an optical fibre from its

optimum shape dramatically reduces the fibre bandwidth, A deviation

from the optimum index profile is caused by the depression in the index

distribution on the fibre axis due to the evaporation of the dopant

material from the innermost layers during the collapsing process.

Linear variation of the refractive index distribution in the dip region

was found to affect pulse dispersion, and the deeper the dip, the

higher the dispersion.

- 136 -

Suggestions for further studies

l. The presence of small ellipticity in the fibre core was found

to improve the launching efficiency of multimode parabolic

index optical fibres. An optimum value of ellipticity could

be found that improves the launching efficiency without

affecting the bandwidth of the fibre. This is of practical

importance since the optimum profile for minimum pulse dispersion

is near parabolic.

2. The improvement of the launching efficiency in non-circular

parabolic index optical fibres, leads to the conclusion that

the ratio of leaky to bound rays is reduced. This in turn

means that there may be a certain value of ellipticity for which

no leaky modes can propagate in the fibres, so that for the near

field scanning profile measurement a reduced correction factor

may be needed for elliptical fibres with a = 2. Experimental

evidence indicates that the full circular correction factor is

required, and the resulting inconsistency could be associated

with the presence of the dip at the core centre,

of which has not been taken into consideration.

the effect

3. It has been shown that a dip at the core centre affects pulse

dispersion and the higher the dip 6, the longer the pulse trail.

- 137 -

In other words, the deeper the dip the higher the dispersion.

This has been investigated assuming that the refractive index

profile in the dip region is linear.

An investigation could be made to determine how an inverse a-profile

refractive index in the dip region affects pulse dispersion.

- 138 -

APPENDICES A - C

- 139 -

APPENDIX A

The power emitted by the source is

p = Jb

r'dr' 0 f2~ f2~ f~/2

dl/J' d$' R'(r',l/J', e• $') sin e'cos e' de' 0 0 0

and the power accepted and guided by the fibre is

=

Ja'r

0dr J2 ~dl/J J

2:$ Je'm R(r ,1/J ,e •• ')T(r ,e')

o o· o o o o o o o 0 0 0 0

sine' case' d6' 0 0 0

(lA)

(2A)

J2~

dl/J 0

0

R(r ,1/J , e• , • )T(r , e• ) sin e• cos e' de' 00 0 0 0 0 0 0 0

.'.n = J: r'dr' J2~ J2~

dl/J1 d•' 0 0

where e' is given by m

and

n' sin e' = n m

n' sin e' = n m

R'(r', 1/J', e', •') sin 8'cose'de'

when considering leaky rays

when considering bound rays only

When the source and the fibre are very close and concentric

r' = • e• o'

= e• 0

- 140 -

fa' f2" f2" Ja ~ r dr d• d• R(r •• ,a•,•) x T(r a•) sine' cos8' de'

0 0 0 0 0 0 0 0' 0 0 0 0 0 0 0 0

R(r ,. ,8' • )sinS' cos8' de' 0 0 0 0

assuming an LED, in which

R' (r', •• , 8' , ••) = R 0 0

Ja'r dr r~ R X T (r , a• l sin a• case' <18' - 0 0 0 0 0 0 0 0 0 0 n =

r r dr 0 0

r/2 R

0 sine' cos'd8'

0 0

assuming T(ro,8' 0 ) = To

2 T r [ sin2 0

a' J r dr f1 = b2 m o 0

0

When considering all rays (bound + skew)

sin a' = l [ n~(ro) r - n22 m I

cos2•

n l - (-E. )

a

and for bound rays only

sin a' = l [n2(ro) 2 1% -, nz m n

2 l - n2

2 2 ) cos •

h =

r

r dr for all rays 0 0

(4A)

(SA)

(6A) .

(7A)

(SA)

and

r 0

- 141 -

r dr 0 0

for bound rays (9A)

- 142 -

APPENDIX B

It was shown in equation (2.22) that the launching efficiency is given by:

n'=

for £ = 0, we have a lambert ion source

n'

i) for b ~ a

= 2 T

0

Taking the refractive index variation as

= n 0

2 [1 - 21\(~)Cl J

Substituting this into equation (lB)

n I =

=

after some

n =

2 T 0

2 T 0

2 T n 0 0

b2

2 r ~21\ 0

manipulation this

4 T n 21\ (!)2 [t 0 0 b

J 21\(!.)Cl a

gives:

Cl ~ 2 J

r dr 0

r dr 0 0

0

r dr 0 0

r dr 0 0

(lB)

(2B)

(3B)

(4B)

(5B)

- 143 -

ii) b < a

In this region the acceptance of the fibre is given by (g),

sin e a (n2(r) 2)~ = n2 m b

2 T r [ :: (n2(r)- n/) 1

0 r dr n =

2 0 0 a 0

assuming the same refractive index variation as before

I

n

=

='Hn!::. --·- 2 [1 o 0 2

r exj - 2ll(a) rdr.

[ 26 - 2ll(~)ex J rdr

1

- _1_ (£_a) ex X b2J. ex + 2

ex + 2

(6B)

(7B)

(8B)

(9B)

(lOB)

- 144 -

APPENDIX C

The scalar wave equation in elliptical co-ordinates can be represented Q...

in the sepAable form given by equations (2.39) and (2.40) where

(lC)

For parabt!lic index elliptical fibre {a = 2), g(e:)is given by equation (2.33)

i.e.

(2C)

Substituting (2C) in lC gives

G(c) 2 2 = e r [

2 '2 2 (n0

- a )sinh E - (n 2 0

2 2 . 2 2 ] 2 2 2' - n2 )e s1nh e:cosh £ -(n0 -n2

)r ~ (3C)

= ·2 2<· 2 ·2 2 4 2 2 2 2 e r n0

- a )sinh E - e r (n0

- n2

)sinh E 4 2( 2 e r n o.

(4C)

2) . h4 n2 s1n e:

(SC)

An E-type mode in elliptical fibres is confined to the region (23)

and

where y 2

£, <e: <e: m1n tp

G(o) = 0 at E = e: tp and e: = e: •

ffi10

= (n2-n22) 0

2 2 ·2 J . h2 2 2' e r (n0 2 - a ) Sln Etp + y r ~ = 0 (6C)

1 -22

y e (n 2

0

- 145 -

. h2 Sln E + tp

11 4 = 0 (7C) e

By completing the square, factorizing and re-arranging:

2sinh 2 £

tp

2sinh 2£ tp

Similarly:

(n 2 0

= 1 ( 2 - 8·2) e2Y2 no

(n 2 0

(n 2 0

(8C)

•2 411 }

2 A]~ - 8 ) - 1 - e

4 (9C)

< 10e.>

- 146 -

APPENDIX D

1. Field Equations:

The well known Maxwell's equations are

.. .. V X E = -jw ll H

0 (lD)

.. .. V X H = j ££0 E (2D)

.. .. E and H are the electric and magnetic field vectors respectively.

= frequency of radiation

£0

(1J0

)= permittivity (permeability) of vacuum

£ = permittivity of propagation medium

j(wt - mA - az) Assuming the dependence e "' where m is an integer

and represents the azimuthal mode number.

B is the propagation constant and t represents the time variation

Using cylindrical co-ordinates system where r, ~· z, denote the radial,

angular; imd axial co-ordinates and through the use of the curl operator,

Maxwell's equations become

1 3 jm j <il<o £ E = ( r H ~) + H z r ar r r (3D)

iw£0 £E a = - j aH - ~ H r z (4D)

"wto·£E = jBH~ - i.!1! H J r r z (SD)

- 14 7 -

(60)

( 70)

- jw~ H = j8 E - ~ E o r ~ r z (SD)

By subsequent substitutions, it is possible to obtain a coupled second

order differential equation for Ez and Hz.

and for H

11 E

z

z

H" z + [~-

( 2£' 8822)]

e: k e: 0

=

e:'k2

1 0

2 - 82) e:(k £ 0

=

H' z

jw ~Jrn.e:'

e:r(k2e: - 82) 0

+ (k2£ 0

2 - 8 -

H z

m

r

E z

2 m

2 ) Ez r

2 ) H

2 z

where the prime denotes differentiation with respect to r.

The coupling term on the right hand side of equations(9D)and(lOD)

vanishes if m= 0 or if the permittivity is constant.

(90)

(loo)

- 148 -

For n = 0, there is no angular variation and both TE and TM modes exist

since Ez = 0 does not imply that Hz = 0. It is possible to eliminate

one of the axial field components E or H to obtain a linear homogeneous z z

fourth order differential equation which is cumbersome to handle. An

approximate solution could be obtained to equations (90) and (lOD) for

modes whose fields are largely confined to the region of maximum permit-

tivity at the core centre. This approximate solution is described in

Reference (S) of Chapter 2.

- 149 -

2. Bessel Functions

The differential equation

dw + z­

dz + <i + ,}, = 0 (UD)

will accept as solutions the linearly independent combination of

J (z) and Y {z) V V

while the equation

+ z dw dz = 0

have as solutions combinations of I (z) and K {z) V V

where

J = Bessel function (1st kind)

y = Bessel function (2nd kind)

I = modified Bessel function {lst kind)(modified Hankel

K = modified Bessel function (2nd kind) (modified Hankel

(120)

function)

function)

In the region where z is real and v is an integer, J and Y exhibit

an oscillatory behaviour.

For a fixed order (v), K decreases with increasing argument while

I increases with increasing argument.

- 150 -

These functions have a series represenation of the form

y {z) 2 = m 11

rlin(~) - - 2

- ljl(m + 1)]

m! <fl

2 - {-) 11

11

-m

"' "'-.,.

L_ k = 1

J {z) m

m - 1 z k ~- <z-l Jk(z)

L_ (m - k)k! k = 0

(-I)k (m+ 2k)

k(m + k)

The subscript m indicates that the validity of the series is

limited to integer orders. ljJ is known as the logarithmic

derivative of the Gamma function

ljl(m) =

ljJ( m) =

d dz

m -- 1 ""'". - Eu + L:_ --k = 1

where Eu is Euler 1 s constant.

-1 k

= r• (m)

r{m)

Similarly_ I and K are represented by the following series:

2 k "' (~- )

(z) {~)V ~ 4 I = ./ V 2

k = 0 k! r< v + k + I)

(130)

(140)

(150)

(160)

(170)

K (z) m

= !. (~) 2 2

- 151 -

-m m - l ~ L_

(m - k - 1)' 2 k ..>.:::._.;;_,.__:~· (- ~)

k = 0 k! 4

+ (-l)m + 1 lin(~) I (z) 2 m

m l z + (-1) - (-) 2 2

00

2 { ,P(k

k = 0

where .pis as defined by equation (15D).

(18D)

- 152 -

3. Gamma Function

This is given by

r(z) = r tz-1 e -t

dt R(z) > 0 (190) 0

which is known as Euler's Integral.

It can also be expressed as (2)

m

r(z) ~ k = 0 z + k

= 1 -z-1 t dt + (200)

The representation given by (200) allows for negative values of the

argument as well as positive.

4. Beta Function

is defined as:

B (z,w) + w

= dt

0

=

and can be expressed in terms of Gamma function as

B(z,w) = r(z)r(w)

r(z + w)

(210)

(220)

- 153 -

S. Incomplete Beta Function

or

B (a,b) X

I (a,b) X

=

=

=

l

r 0

ta- l (l - t)b - l dt

Bx(a,b) / B(a,b)

I (a,b) X B(a,b) I

X ta - l (l - t)b - l dt

0

where B(a,b) is the Beta function

and in a series form, it can be represented as

= 2 n = 0

B(a + 1, n + l) B(a + b, n + 1)

(23D)

(24D)

(2SD)

x n+l 1 (260)

- 154 -

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- 156 -

CHAPTER 2

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pp. 369-386, 1974.

7. A.W. Snyder and D.J. Mitchell, 'Leaky rays on circular optical fibres',

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Peregrinus, pp. 169-177.

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11. M. Born and E. Wolf, 'Principles of optics', Pergamon Press, Oxford,l964

- 157 -

12. D.L. Bisbee, 'Optical fibre joining technique', B.S.T.J., Vol. SO, No. 10,

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- 158 -

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- 159 -

CHAPTER 3

l. J. B. Macchesney, P. B. O'Connor and H.M. Presby, 'A New technique

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- 160 -

10. J.A. Arnaud, 'Beam and fibre optics', New York, Academic Press, 1976.

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- 161 -

CHAPTER 4

1. W.Eickoff and 0. Krumpholz, 'Determination of the ellipticity

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Elec. Lett. Vol. 12, No. 16, pp. 405-407, 1976.

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1971.

4. V. Ramaswamy, R. Standly, D. Sze and W. French, 'Polarization effects

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Progress in pptical fibre communication, Vol. 2, by P.J.B. Clarricoates,

IEE reprint series, pp. 23-24, Peter Peregrinus, 1982

- 162 -

10. V. Ramaswamy, W. French and R. Standly, 'Polarization characteristics

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1978 .•

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1.5 ~m wavelength region', Elec. Lett. 15, pp 476-478, 1979.

12. A.M. Smith, 'Polarization and magneto -optic properties of single

mode optical fibre', Appl. Optic. 17, pp. 52·56, 1978.

13. S. Norman, D. Payne, M. Adams and A. Smith, 'Fabrication of single

mode fibres exhibiting extremely low polarization birefringence',

Elec. Lett. 15, pp. 309-311, 1979.

14. C. Yeh, 'Elliptical dielectric waveguides', J. Appl. Phys. 33,

pp. 3235-3243, 1962

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CHAPTER 5

1. D. Gloge and E. Marcatili, 'Multimode theory of graded-core

fibres', BSTJ Vol. 52 No. 9, pp 1563-1579, 1973.

2. R. Olshansky and D. Keck, 'Pulse broadening in graded index

optical fibres', Appl. Optic., Vol. 15, No. 2, pp 483-4'91, 1976.

3. D. Marcuse and H. Presby, 'Effect of profile deformations on

fibre bandwidth', Appl. Optic. Vol. 18, No. 22, pp. 3758-3763, 1979.

4. R. Hanson and E. Nico1aisen, 'Propagation in graded index fibres:.

Comparison between experiment and three theories 1 , Appl. Optics

17, pp 2831-2835, 1978.

5. R. Hensen, 'Pulse broadening in near optimum graded index fibres',

Opt. and Quan. Elec. 10, pp. 521-526, 1978.

6. D. Marcuse. 'Calculation of bandwidth from index profiles of

optical fibres 1: Theory', Appl. Optic. 18. pp. 2073-2080, 1979.

7. P.L. Chu, 'Calculation of impulse response of multimode optical

fibres', Progress in Optical Communication by P.J.B. Clarricoates,

vol. 2. IEE reprint series, Peter Peregrinus, pp. 17-19, 1982.

8. E. Marcatili, 'Modal dispersion in optical fibres with arbitrary

numerical aperture and profile dispersion', B.S.T.J. 56, No. 1,

pp. 49-63, 1977.

9. W. Streiter and C. Kurtz, 'Scalar analysis of radially inhomogeneous

guiding media', J. Opt. Soc. Am. 57, pp. 779-786, 1967.

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10. K. Kitayama, S. Seikai and N. Uchida, 'Impulse response prediction

based on experimental mode coupling coefficient in a 10 km grade

index fibre', IEEE J. Quan. Elec. Vol. QE-16, No. 3, pp 356-362,

1980.

11. H. Presby, D. Marcuse and L. Cohen, 'Calculation of bandwidth

from index profiles of optical fibres', Appl. Opt. Vol. 18, No. 19,

pp. 3249-3255. 1979.

12. D. Marcuse, 'The impulse response of an optical fibre with

parabolic index profile', BSTJ Vol. 52, No. 7, pp. 1169-1174,

1973.

13. D. Gloge and E. Marcatili, 'Impulse response of fibres with ring­

shaped parabolic index distribution', ibid Vol. 52, No. 7,

pp. 1161-1168, 1973.

14. R. Olshansky, 'Pulse broadening caused by deviation from optimal

index profile', Appl. Opt. Vol. 15, No. 3, pp. 782-788, 1976.

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CHAPTER 6

1. D.L. Bisbee, 'Optical fibre joining technique',

BSTJ, Vol. 50, No. 10, pp 3153-3158, 1971.

2. D. Gloge, P. Smith, D. Bisbee and E. Chinnock, 'Optical fibre

and preparation for low-loss splices',

BSJT, Vol. 52, No. 9, pp 1579-1588, 1973.

3. H. Eyyuboglu, 'Attenuation of leaky modes and filtering techniques',

Ph.D. Thesis, L.U.T., 1981.

4.- W.B. Allen, 'Fibre optics, theory and practice', Plenum Press,

1973.

5. R.S. Longhurst, 'Geometrical and physical optics', Longman Press,

1973.

. 6. C.M. Miller 'Transmission vs. Trcnsverse offset for parabolic

profile fibre splices with unequal core diameter',

BSTJ, Vol. 55, No. 7, pp 917-928, 1976.

7. M. Young, 'Geometrical theory of multimode optical fibre to fibre

connectors', Optic. Cornrn. No. 3, pp 253-255, 1973.

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APPENDIX D

1. C. Kurtz and W. Streifer, 'Guided waves in inhomogeneous

focusing media, Part 1: Formulation', Optical fibre waveguides

by P.J.B. Clarricoates, lEE reprint series, Peter Peregrinus,

pp 10-14, 1975.

2. G. N. Watson, 'Theory of Bessel Functions', Cambridge University

Press, Cambridge, 1958.

3. M. Abramowitz and I. Stegun, 'Handbook of Mathematical Functions',

Dover publications, New York, 1968.

4. N. Lebedev, 'Special functions and their applications', Prentice­

Hall, New Jersey, 1965.

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