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Waveguide propagation of microwave signals at frequencies less than cut-off

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ACADEMU' U E G i ^ n u n .
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PAY N E,V B .
WAUEGUIDE
P
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’’W aveguide P ro p a g a tio n o f M icrowave S ig n a ls
a t P re q u e n o ie s l e s s th a n C u t-O ff”
A t h e s i s p r e s e n te d f o r th e
d egree o f D o c to r o f P h ilo so p h y
i n th e U n iv e r s ity o f London.
by
V a le r ie Beynon P ayne, B .S o ., M*8 o . , A .R .C .S .
J u l y , 19680
D epartm ent o f P h y s ic s ,
U n i v e r s i ty o f S u rre y ,
London, 8,W ,11,
-2 -
AB8TBACT.
U sin g tra n s fo rm te c h n iq u e s , a tim e d ependent s tu d y h a s b e en
made o f th e e v a n e s c e n t wave i n a w aveguide o r s i m i l a r d i s p e r s iv e
medium,
F i r s t l y , th e f i e l d c r e a te d by th e a p p l i c a t i o n o f a u n i t s te p
m o d u lated c a r r i e r wave to an u n te r m in a te d w av eg u id e, c u t - o f f T /ith
r e s p e c t to th e c a r r i e r fre q u e n c y , h a s b een i n v e s t i g a t e d a s a
f u n c t i o n o f tim e and t h e d is ta n c e from th e p o in t o f a p p l i c a t i o n .
The b e h a v io u r o f th e t r a n s v e r s e e l e c t r i c and m a g n e tic f i e l d
com ponents and th e r e s u l t i n g P o y n tin g v e c t o r h a s
b e e n p r e s e n te d
g ra p h ic a lly .
Secondly, th e i n t e r a c t i o n o f nanosecond m icrowave c a r r i e r
p u ls e s w ith a te r m in a te d s e c t i o n o f c u t - o f f w aveguide was
i n v e s t i g a t e d t h e o r e t i c a l l y and e x p e r im e n ta lly ,
T h is w aveguide
was c u t - o f f w ith r e s p e c t to th e c a r r i e r fre q u e n c y and was
in te r p o s e d b etw een two i d e n t i c a l p ro p a g a tin g g u id e s .
U nder
c e r t a i n c o n d itio n s i t i s p o s s ib le to p r e d i c t r e f l e c t i o n and
tr a n s m is s io n tim e s f o r th e p u ls e on i n t e r a c t i o n w ith th e c u t - o f f
guide»
These r e f l e c t i o n i n t e r a c t i o n tim e s have b e e n c o r r e l a t e d
w ith th o s e m easured e x p e r im e n ta lly .
T h is tim e d ep en d e n t stu d y
form s an e x te n s io n o f th e s te a d y s t a t e microwave a n alo g u e o f th e
quantum m ech an ica l tu n n e l e f f e c t .
-3 CONTENTS
T i t l e page
1
A b s tr a c t
2
C o n te n ts
3
N o ta tio n and Num bering o f s e c t i o n s , e q u a tio n s and f i g u r e s
8
CHA-FTER 1 .
S e c tio n
INTRODUCTION TO AND AIMS OR THE INVESTIGATION.
1,1 The e v a n e s c e n t wave : d e f i n i t i o n and Im portance
i n th e s te a d y s t a t e ,
13
1 .2 The e v a n e sc e n t v/ave: th e tim e d e p en d en t problem
and aim o f t h i s i n v e s t i g a t i o n ,
1 .3 G e n e ra l tim e d e p en d e n t c o n s id e r a tio n s and v e l o c i t i e s ,
19
20
1,A S p e c if ic tim e d e p en d e n t c o n s id e r a tio n s and th e form
o f th is in v e s tig a tio n ,
CHAPTER 2,
21
A STUDÏ OF THE PROPAGATION OF A UNIT STEP MODULATED
CARRIER WAVE IN A WAVEGUIDE,
S e c tio n
2 ,0 I n t r o d u c t i o n
2.1 I n t e g r a l tr a n s fo r m te c h n iq u e s
2 .2 The L a p la c e tra n s fo rm
2 .3 L a p la c e tra n s fo r m te c h n iq u e s a p p lie d to M axwell*s
e q u a tio n s
2A
2A
25
27
2 ,if R e s t r i c t i o n s on th e s tra n s fo rm e d i n i t i a l c o n d itio n s
i n th e tim e dom ain,
2 .5 T y p ic a l i n i t i a l tim e c o n d itio n s and t h e i r s tra n s fo r m s
29
30
2 .6 The u se o f t a b u l a t e d i n v e r s io n i n t e g r a l s and th e
c o n v o lu tio n th eo rem
30
•2 ,7 D i r e c t e v a lu a tio n o f th e i n v e r s io n i n t e g r a l i n th e
com plex fre q u e n c y p la n e f o r o < t < z / c .
33
-4 -
2*8
D ir e c t e v a l u a t i o n o f th e i n v e r s io n i n t e g r a l i n th e
complex fre q u e n c y p la n e f o r t > z / o *
2 .9
36
S addle p o i n t m ethod o f i n t e g r a t i o n , t > z / c , a sy m p to tic
s o l u t io n .
38
2.1 0 E v a lu a tio n o f th e i n v e r s io n i n t e g r a l f o r sm a ll v a lu e s
o f ( t ~ z /o ).
li-1
2.11 E v a lu a tio n o f th e a p p ro x im ate s o l u t io n s o b ta in e d th u s
f a r and th e n e e d f o r an e x a c t s o l u t io n ,
2 .1 2 E x act e v a l u a t i o n o f th e i n v e r s i o n i n t e g r a l i n th e
if3
2f3
complex fre q u e n c y p la n e f o r v a lu e s o f tim e , t > z / c .
2 .1 3 The com plete s o l u t i o n f o r th e tr a n s v e r s e e l e c t r i c and
m ag n etic f i e l d s .
2 .1 4 N u m erical c o m p u tatio n s o f th e e x a c t and ap p ro x im ate
fu n c tio n s .
34
2 .1 3 V a r ia t io n o f t h e tr a n s v e r s e f i e l d com ponents w ith tim e
(lO^t) , f o r g iv e n v a lu e s o f ( f ^ / f ^ ) and d is ta n c e
(z / X j
2o16 The I n s ta n ta n e o u s P o y n tin g v e c t o r
61
6j
2 .1 7 V a r ia t io n o f th e t r a n s v e r s e f i e l d com ponents w ith
d is ta n c e ( z / \ ^ ) f o r g iv e n v a lu e s o f ( f ^ / f ^ ) and
tim e , ( 4t)^t) ,
2 .1 8 D is c u s s io n and C o n c lu sio n s.
73
78
CHAPTER 3 . A TIME DEPENDENT STUDY OP "PROSTRATED TOTAL REELECTION",
S e c tio n 3»0
In tro d u c tio n
83
3.1
The ^tu n n e l e f f e c t* - s te a d y s t a t e c o n s i d e r a ti o n .
83
3 .2
The *t u n n e l e ff e c t* - th e tim e d ep en d en t p ro b lem ,
84
-5 ■►
3*3
The S t a t i o n a i y P hase M ethod,
3 .4
The w aveguide ' t u n n e l e ff e c t* - s te a d y s t a t e
c o n s id e r a tio n s ^
3 .3
93
G e n era l d i s c u s s i o n o f i n t e r a c t i o n tim e s f o r d i s t o r t e d
p u ls e s .
3 .8
P r a c t i c a l a p p l i c a t i o n o f th e a n a l y s i s ,
3 .9
F o o tn o te on th e quantum m ec h an ica l t u n n e l e f f e c t ,
CHAPTER 4 .0
90
P r e d i c t i o n o f group r e f l e c t i o n and tr a n s m is s io n
i n t e r a c t i o n tim e s u n d e r c e r t a i n c irc u m s ta n c e s .
3 .7
88
The S ta tio n a z y p h a se m ethod a p p lie d to th e w aveguide
tu n n e l e f f e c t ,
3 .6
87
97
93
101
EXEERIMEETALl BTVESTIGATIGNS WITH NANOSECOND MICROWAVE
CARRIER PULSES
S e c tio n 4 .0
P a rt I .
In tro d u c tio n
103
P r e lim in a r y m easurem ents u s in g nanosecond microwave
c a r r i e r p u ls e s
4 .1
G e n e ra tio n and m easurem ent te c h n iq u e s o f nanoseco n d
microwave p u l s e s ,
t03
4 .2
P re v io u s p u ls e i n v e s t i g a t i o n s i n w a v eg u id e .
103
4 .3
The aim o f th e e x p e rim e n ts .
IO6
4 .4
P ro d u c tio n o f th e n anosecond m icrowave p u l s e s .
109
P a rt I I ,
The S tep p ed W aveguide T e s t S e c tio n .
4*3
R e f l e c t i o n i n t e r a c t i o n ex p erim en t w ith lo n g g u id e s
II3
4 .6
E x p e rim e n ta l P ro c ed u re
117
4 .7
D is c u s s io n o f th e r e f l e c t e d p u ls e re c o r d in g s and
m easurem ents p e rfo rm e d on them ,
119
Zj.,8
C o n tro l e x p e rim e n t f o r r e f l e c t i o n from th e lo n g s te p p e d
g u id e u n i t ,
if ,9
126
E v a lu a tio n o f th e m ea su rin g system and r e s u l t s o b ta in e d , 126
if, 10 F u r th e r r e f l e c t i o n i n t e r a c t i o n e x p e rim e n ts w ith lo n g and
s h o r t m o d ifie d g u id e s ,
I 36
4.11 Q u a l i t a t i v e E v a lu a tio n o f th e r e s u l t s o f th e s e f u r t h e r
expe lim e n ts .
139
4 .1 2 T ransm ission experim en ts Tfith sh ort step p ed waveguide
s e c tio n s .
144
4 .1 3 E v a lu a tio n o f th e tr a n s m is s io n e x p e rim e n ts,
P a rt I I I
143
The d i e l e c t r i c f i l l e d w aveguide t e s t s e c t i o n R e f le c tio n and T ra n sm iss io n E x p e rim e n ts,
4 .1 4 I n t r o d u c tio n „
-) 32
4 .1 5 D e s c r ip tio n o f th e t e s t s e c t i o n .
I 32
4 .1 6 E x p e rim e n ta l P ro c e d u re f o r th e R e f le c t i o n E x p e rim e n ts,
4 .1 7 E v a lu a tio n o f th e m e a su rin g
4 .1 8 T ra n s m is s io n e x p e rim e n ts
system and r e s u l t s o b ta in e d . 139
w ith s h o r t u n f i l l e d s e c t i o n s ,
4 .1 9 E x p e rim e n ta l P ro c e d u re ,
system and th e r e c o r d in g s
o f th e t r a n s m i t t e d p u l s e s .
165
4.21 C oncluding Rem arks,
SÜMiARÏ MD CONCLUSIONS.
S e c tio n 3 .0
In tro d u c tio n
167
I 68
3.1
The tim e d e p en d e n t s tu d y
of
th e e v a n e sc e n t w ave,
3 .2
The tim e d ependent s tu d y
of
" F r u s t r a t e d T o ta l
R e fle c tio n " .
162
162
4 .2 0 E v a lu a tio n o f th e m easu rin g
CHAPTER 3
32^
I 69
172
7-
APPENDIX A.
System o f t r a n s v e r s e m agnetic and tr a n s v e r s e e l e c t r i c
d ire c t
APPENDIX B,
waves i n th e com plex f re q u e n c y , s , p la n e ,
R e s t r i c t i o n s on th e a n a l y t i c a l form
o f th e s tra n s fo rm e d
178
184
i n i t i a l c o n d it i o n s , A ^ ( o ,s ) , A^^^^ ( o , s ) , B ^ (o ,s ) and
EEPEEENCES.
18?
ACXÏÏ0W1ED&EMENTS.
I 90
NOTATION
The fo llo w in g i s a l i s t o f sym bols and t h e i r commonly u se d
m ean in g s.
Some o f th e sym bols have a ls o b een u se d i n v a r io u s
s e c t i o n s a s p a ra m e te rs i n m a th e m a tic a l e x p re s s io n s b u t such l i m i t e d
u se i s o b v io u s from th e c o n te x t.
C e r ta in sym bols a re fo llo w e d by
a l i s t o f th e s u b s c r i p t s w hich th e y ta k e ,
CC
(o) r e a l p ro p a g a tio n c o n s ta n t
(3
( 0 , 1 , 2) j p , im a g in a l y p r o p a g a tio n c o n s ta n t
y
( 1 , 2) com plex p r o p a g a tio n c o n s ta n t = a + j p
l/Y o
= % ± j (1
e
p e rm ittiv ity
-
r e la tiv e p e rm ittiv ity
p e rm ittiv ity o f fre e space, 1
*9 *10^)
f a r a d s /m e tr e
©n(^^ >^2)
( ^ '^ ^ ^ ^ t h a t p a r t o f th e in s ta n ta n e o u s m a g n e tic f i e l d
component w hich i s a f u n c tio n o f th e tr a n s v e r s e
/<
A
c o - o r d i n a te s , x . , x _ , o n ly
% w ./C
'
w a v e le n g th
(% 2) c u t - o f f w a v e le n g th
\i
p e r m e a b ility
p e r m e a b ility o f f r e e s p a c e ,
^
X
3.14159
w t
c
4)
( R ,T ,0 ,2 ) phase f u n c t i o n
w
a n g u la r fre q u e n c y ,
2 'rrf
h e n r ie s /m e tr e
•9"
w
a n g u la r c a r r i e r fre q u e n c y
( 1 , 2) a n g u la r c u t - o f f fre q u e n c y , w
4»
= ^ /(
±
^
wave f u n c tio n
^
( 1 , 2 , 3) t h a t p a r t
o f th e in s ta n ta n e o u s e l e c t r i c f i e l d
com ponent, w hich i s a f u n c t i o n o f th e t r a n s v e r s e
c o - o r d i n a te s ,
x .,
o n ly
( 1 , 2 , 3) t h a t p a r t o f th e in s ta n ta n e o u s e l e c t r i c / m a g n e t i c
n ( ^ f i e l d
com ponent, w hich i s a f u n c t i o n o f th e q u a n t i t i e s
z and t
A ^ ( z ,s )
,
^
o n ly .
( 1 , 2, 3) s - tra n s fo rm e d f u n c t i o n
> 4 ^ ( z ,t )
th e d e r i v a t i v e w ith r e s p e c t to z
o f A ( z ,s )
B ^ (z ,s )
a
le n g t h o f c u t - o f f s e c t i o n
B/A
r e f l e c t i o n c o e f f i c i e n t w ith
Ib/AI
,a )
b
b
m odulus and
phase
b ro a d d im en sio n i n r e c t a n g u l a r w aveguide
t
c/A
1C/A I
re d u c e d b ro a d dim ension
tr a n s m is s io n c o e f f i c i e n t
vfith
m odulus and
o( CÜ,a )
phase
0
v e l o c i t y o f l i g h t i n medium, 1 / ( p c ) ^
0^
v e lo c ity o f l ig h t
i n f r e e sp a c e ,
c o s ((2 n + 1 ) 8 ) , 6 = a rc c o s ( v ^)
m e tr e s /s e c
1
'1 0 -
= cosh ( ( 2 n + l ) a ) ^
a = a rc cosh ( v
^ 1
e n e rg y o f p a r t i c l e ( quantum m echanics)
c
C
H
I n s ta n ta n e o u s e le o tr io /m a g n e tio f i e l d v e c to r s
( 1 , 2 , 3) com ponents o f th e in s ta n ta n e o u s e l e c t r i c
^
m a g n e tic f i e l d v e c t o r s
( x ^ , Xg)
®n ( ^ 1 '
^ n (z )
n
d e r i v a t i v e , v /ith r e s p e c t to z , o f
£n
M i(z )
L a p la c e tra n s fo rm e d ( w ith r e s p e c t t o s) f i e l d
«n
dom inant mode i n r e c t a n g u l a r w aveguide
E ( s)
s
- tra n s fo rm e d i n i t i a l c o n d itio n i n th e tim e
domain a t z = o
^ q( s)
p ro to ty p e s - tr a n s f o r m e d f i e l d component
E n (s)
( 1 , 2 , 3 , 4 ) o t h e r s tra n s fo rm e d f i e l d com ponents
f
fre q u e n c y
f^
c a r r i e r fre q u e n c y
f^
c u t - o f f fre q u e n c y
f ( o ,t)
i n i t i a l c o n d it i o n i n th e tim e domain a t z=o
f ( z ,t)
( n = o ,2) e x a c t s o l u t i o n s f o r th e in s ta n ta n e o u s
f n ( 2, t )
f i e l d com ponents
:^ ( z , t)
( 0 , 2) ap p ro x im ate s o lu tio n s f o r th e in s ta n ta n e o u s
f i e l d com ponents
( 1 , 2)
( 1, 2^3)metrio c o e f f i c i e n t s
j
(-1 )"
k
wavenum ber, 2 r f / X
( 1 , 2) c u t - o f f wavenuniber
X,
L a p la c e tr a n s f o r m a tio n
-1
I n v e r s e L a p la c e tr a n s f o r m a tio n
p
n o rm a liz e d fre q u e n c y , s / w ^
Pg
sa d d le p o in t fre q u e n c y
P
in s ta n ta n e o u s P o y n tin g v e c t o r ,
r e f e r r e d to a s th e
P o y n tin g v e c t o r
P
tim e a v e ra g e P o y n tin g v e c t o r
8
complex a n g u la r fre q u e n c y
t
tim e
tj^
r e f l e c t i o n i n t e r a c t i o n tim e
t^
tr a n s m is s io n i n t e r a c t i o n tim e
V (z)
p o t e n t i a l en erg y f u n c t i o n (quantum m ech an ics)
V*^
p o t e n t i a l en erg y o f b a r r i e r r e g io n
V
6
W(p, % )
group v e l o c i t y
th e e x p o n e n t,
=
T: ) =
x^
X
px
-
/< (p
pT
2
9 ^ ( w ( p , t ))
a 2
3P
-
(ÿ) (p)
“
+ 1)2
f o r w aveguide o r plasm a
P = Pg
( 1 , 2) t r a n s v e r s e c o - o r d in a te s
tra n sv e r se c o -o r d in a te s i n th e r e c ta n g u la r system
y
l o n g i t u d i n a l c o - o r d in a te
.12^
l(t-a )
u n i t s te p f u n c t i o n
s o fo r t < a
= 1 fo r t > a
Num bering o f s e c t i o n s , e q u a tio n s and f i g u r e s
Each s e c t i o n h a s b een s p e c i f i e d by a two num ber in d ic a to r *
T h u s, (2,2{-) i n d i c a t e s th e f o u r t h s e c t i o n o f C h a p te r 2«
The
e q u a tio n s o f a p a r t i c u l a r s e c t i o n a r e s p e c i f i e d by a t h r e e number
in d ic a to r.
( 2, 4) .
T hus, (2,4<»6) i n d i c a t e s th e s i x t h e q u a tio n o f s e c t i o n
Since th e s e c t i o n i n d i c a t o r h a s been ty p e d i n th e to p r i g h t
h an d c o rn e r o f e ac h p a g e , t h i s m ethod o f num bering f a c i l i t a t e s b a c k
r e f e r e n c e to e q u a tio n s ,
P ig u re s have b e e n num bered i n th e o r d e r i n w hich th e y a p p e a r.
How ever, when m en tio n ed i n th e t e x t , t h e i r
in c lu d e d i n b r a c k e t s a f t e r
page
th e f i g u r e num ber.
num bers have b een
-1 3 -
(1.1)
CP-EPER 1 .
IHTRODUCTION TO iNP AIMS OF THE mVESTIGATION
1.1
The ev aziescent wave;
d e f i n i t i o n and im p o rta n c e i n th e s te a d y
s ta te .
The e v a n e sc e n t e le c tro m a g n e tic wave o r l i t e r a l l y , th e wave
w hich i s “dying away” h a s i n t e r e s t e d s c i e n t i s t s from b o th an
academ ic and p r a c t i c a l p o in t o f v iew .
A t o p t i c a l f r e q u e n c i e s , an
e v a n e s c e n t wave i s p r e s e n t i n th e d i e l e o t r i o medium o f lo w e r
r e f r a c t i v e in d e x when a l i g h t beam, i n c i d e n t i n a medium o f h ig h e r
r e f r a c t i v e in d e x , i s t o t a l l y r e f l e c t e d a t th e i n t e r f a c e o f th e
two d i e l e c t r i c m ed ia.
The a m p litu d e o f th e wave i n th e r a r e r
medium d ecay s e x p o n e n tia lly w ith d is ta n c e away from th e i n t e r f a c e .
The r e f l e c t i o n i s t o t a l s in c e , a lth o u g h th e r a r e r medium s u p p o rts
t h i s e v a n e sc e n t v/ave, no en erg y i s p ro p a g a te d i n t o th e second
medium i n th e d i r e c t i o n norm al to th e i n t e r f a c e b u t i s c o m p le te ly
re fle c te d .
An e v a n e sc e n t wave may a ls o b e su p p o rte d i n a w aveguide o r
h o llo w m e ta l c y l i n d e r .
The p re s e n c e o f th e m e ta l w a lls n e c e s s i t a t e s
th e d e s ig n a tio n o f “ c u t - o f f ” f r e q u e n c i e s , f ^ .
For p e rfe c tly
c o n d u c tin g w a l l s , th e p ro p a g a tio n c o n s ta n t, Y , o f waves o f
f re q u e n c y , f , i s g iv e n by
Y =
2 rr(f^ - f^ ) ^ /
0
where c = 1 /
F o r f r e q u e n c ie s , f , g r e a t e r th a n f ^
im a g in a ry , j p
^
th e p r o p a g a tio n c o n s ta n t i s
, w h ile f o r f r e q u e n c ie s l e s s th a n t h i s ,
i s r e a l , cc , and th e wave i s e v a n e s c e n t.
it
A g ain , th e a m p litu d e o f
t h i s wave decays e x p o n e n tia lly w ith d i s t a n c e from th e j u n c t i o n
b etw een th e p ro p a g a tin g and “ c u t - o f f ” g u id e s .
( 1 , 1)
*«i 2|~.
F ig u re 1(15 ) i l l u s t r a t e s d ia g ra m m a tic a lly th e c o n d itio n s
p ro d u c in g th e e v a n e sc e n t wave i n th e s e two cases*
The p e n e t r a t i o n o f t h e s e v/aves i n to th e r a r e r medium on t o t a l
Ci")
r e f l e c t i o n was i n v e s t i g a t e d i n 1902 by E .E , H a ll^
The e x p e rim e n ts
in v o lv e d b r in g in g a t h i r d medium to w ith in a s h o r t d is ta n c e o f th e
f i r s t , so t h a t th e r a r e r medium was sandw iched betw een m edia o f
l a r g e r r e f r a c t i v e index*
A p r o g r e s s iv e wave was e s t a b l i s h e d i n th e
t h i r d medium and th e e n e rg y was no lo n g e r t o t a l l y r e f l e c t e d a t th e
f i r s t in te rfa c e .
re fle c tio n " *
T h is i s th e phenomenon o f " f r u s t r a t e d t o t a l
F u r t h e r e x p e rim e n ts to d e m o n stra te t h i s e f f e c t have
b een p e rfo rm e d w ith e le c tr o m a g n e tic waves o f f r e e space w a v e le n g th s ,
15, 3 and 1 . 20ms. ( 2,1 9 1 0 to 1963)
i i g h t ( ^ ’ 1960)^
An e x p la n a tio n o f th e phenomenon can be r e a â ;lly a r r i v e d a t ,
a f t e r th e f o llo w in g c o n s i d e r a ti o n o f th e w aveguide c a s e .
H e re ,
th e phenomenon i s o f p r a c t i c a l i n t e r e s t s in c e i t form s th e b a s i s
o f th e p i s t o n o r c u t - o f f a tte n u a to r ^ ^ ^ an d , more r e c e n t l y , wave­
g u id e band p a s s f i l t e r s u s in g e v a n e sc e n t modes have b e en c o n s tr u c te d,(5)
I n t e r e s t c e n tr e d on t h i s ty p e o f a t t e n u a t o r s in c e i t s a t t e n u a t i o n
c h a r a c t e r i s t i c s c o u ld be c a l c u l a t e d a c c u r a t e l y from a know ledge o f
th e le n g th and o f th e " c u t - o f f " g u id e .
F o r an i n f i n i t e l y lo n g ,
u n te rm in a te d " c u t - o f f " g u id e , th e t r a n s v e r s e e l e c t r i c and m ag n e tic
f i e l d com ponents, C a n d H ^ ^ ,
^ tr
^o
where
-a'Z .) and
a n d li^ a re c o n s t a n t s .
aise i n phase q u a d r a tu r e , t h a t i s ,
exp ( j w t - a z)
T hus, th e e n erg y flo w i n th e
c u t - o f f g u id e , g iv e n by th e tim e a v erag e P o y n tin g v e c to r ( ^ xjHT )
i s z ero .
H ow ever, i f a p r o p a g a tin g g u id e i s i n tr o d u c e d a t a f i n i t e
d is ta n c e a lo n g t h i s g u id e , an e n e rg y flo w can be d e te c te d i n i t .
( 1 . 1)
s i n ( 0 .)
n.
s in ( 9 )
n.
n = r e f r a c t i v e in d e x
s in ( 0 ^ )
=
ng/n^
T o ta l r e f l e c t i o n when
0: > B .
/
rf
/
^c1^ f c 2 ? " c u t o f f f r e q u e n c ie s
■*
^o1
T o ta l r e f l e c t i o n when
^01 < ^ <
\
^o2
^ ^o2 ~
“ ^o2
F ig u re 1 c ,
^
^o1 ~ ^o2
^
^
T o ta l r e f l e c t i o n when
fo 1 <
F ig u re 1 .
o f f w a v e le n g th s
C o n d itio n s
^ < ^02
in g th e e v a n e sc e n t wave f o r
—16*^
(1#1)
The p re s e n c e o f t h i s p r o p a g a tin g s e c t i o n g iv e s r i s e to a r e f l e c t e d
wave i n th e c u t - o f f s e c t i o n and th e s u p e r p o s itio n o f th e i n c i d e n t
and r e f l e c t e d waves to g iv e compound e l e c t r i c and m ag n e tic f i e l d s
le a d s to a non z ero P o y n tin g v e c to r ^ ^ ^ ,
2(17 ) .
T h is i s shown i n f i g u r e
I t T fill he se e n t h a t th e r e s u l t a n t m ag n etic and e l e c t r i c
f i e l d q u a n t i t i e s a r e no lo n g e r i n p h a se q u a d ra tu re an d w i l l have a
tim e a v erag e P o y n tin g v e c t o r , £ îfl cos 0
,
T h is q u a n t i t y i s c o n s ta n t
th ro u g h th e c u t - o f f s e c t i o n h u t f o r a g iv e n r e f l e c t i o n c o e f f i c i e n t
a t th e end o f th e s e c t i o n , i t d ecay s e x p o n e n tia lly w ith th e le n g t h
o f th e s e c t i o n .
The a n a lo g y o f th e phenomenon o f " f r u s t r a t e d t o t a l r e f l e c t i o n "
can a ls o he fo u n d i n th e quantum m e c h a n ic a l p a r t i c l e realm *
H ere i t
i s g iv e n th e name "quantum m e c h a n ic a l tu n n e l e f f e c t ” an d i s e x p re s s e d
i n te rm s o f e n e r g i e s .
I f th e f o r c e f i e l d a c t in g on a p a r t i c l e i s
z ero o r n e a r l y z e ro everyw here e x c e p t i n a l i m i t e d r e g io n , i t i s
s a i d to com prise a p o t e n t i a l b a r r i e r ,
A p a r t i c l e o f e n e rg y , B, i s
i n c i d e n t from th e l e f t on a p o t e n t i a l h a r r i e r i n th e r e g io n z = o
to z = a ,
(se e fig u re 3 (l8 ) ) .
A t z = o i t e n c o u n te rs th e h a r r i e r ,
w hich i s a c l a s s i c a l l y f o rb id d e n r e g io n .
However, th e quantum
m ec h a n ic a l b e h a v io u r o f th e p a r t i c l e shows t h a t i t c a n p e n e t r a t e th e
h a rrie r.
S u p e r p o s itio n o f th e i n c i d e n t and r e f l e c t e d wave f u n c tio n s
i n th e h a r r i e r r e g io n y i e l d s a f i n i t e p r o b a b i l i t y o f th e p a r t i c l e
b e in g t r a n s m i t t e d a t th e o t h e r s id e o f th e h a r r i e r a t z = a .
p r o b a b i l i t y d ecay s e x p o n e n tia lly w ith th e d i s t a n c e , a .
T h is
T h u s, t h e r e
i s a quantum m e c h an ica l p o s s i b i l i t y t h a t th e p a r t i c l e w i l l p e n e t r a t e
o r tu n n e l th ro u g h th e p o t e n t i a l h a r r i e r .
H u p ert
( 7)^ h a s dravm a com plete an alo g y betw een th e w aveguide
( 1 . 1)
•17“
B
2b
AC i s th e ”c u t - o f f ” s e c t i o n
A power flo w e x i s t s to th e r i g h t o f AB.
2 a, 2b«
P r a c t i c a l r e a l i z a t i o n o f " f r u s t r a t e d ”t o t a l
r e f l e c t i o n u s in g a w aveguide system .
wt
H
X
H
■Hi
- i n c i d e n t t r a n s v e r s e e l e c t r i c and
m ag n e tic f i e l d com ponents,
- r e f l e c t e d f i e l d com ponents.
- r e s u l t a n t t r a n s v e r s e f i e l d components
F ig u re 2oc
Time v e c t o r diagram o f th e r e s u l t a n t t r a n s v e r s e
f i e l d com ponents f o r " f r u s t r a t e d t o t a l r e f l e c t i o n ” .
-18*
(iJ)
V( s)
V
E
a
Fipiure 3 a .
R e p r e s e n ta tio n o f th e p o t e n t i a l b a r r i e r
S ta n d in g wave r e g io n
B a r r i e r r e g io n
III
P r o g r e s s iv e wave r e g io n
III
o
F ig u r e 5b»
a
z
P i c t o r i a l R e p r e s e n ta tio n o f th e wave f u n c t i o n o f a
p a f t i c l e , w ith e n erg y
, i n c i d e n t on th e
l a r r i e r r e g io n ( l l ) from th e led
-1 9 -
( 1 .1 )
and quantum m e c h a n ic a l a s p e c ts o f th e s u b je c t and h a s u se d a
microwave e x p e rim e n t to d e m o n s tra te th e " tu n n e lin g e f f e c t " .
S im ply,
he h a s p la c e d a s h o r t c u t - o f f s e c t i o n betw een two s e c tio n s o f
p ro p a g a tin g g u id e ( s e e f i g u r e 2b( 1 7 )) and m easu red th e f i e l d
s t r e n g th i n t r a v e r s i n g th e c u t - o f f s e c tio n from th e f i r s t to th e
second p r o p a g a tin g s e c t i o n s .
The ex p erim en t shows q u ite c l e a r l y
th e e x p o n e n tia l d ecay o f th e e l e c t r i c f i e l d a m p litu d e e x p e c te d
a c r o s s th e c u t - o f f s e c tio n ,
Campi and H a r r i s o n ^ h a v e u se d th e
r e c t a n g u l a r b a r r i e r o f f i g u r e 2a( 17 ) and a ls o a h y p e rb o lic b a r r i e r
i n t h e i r d e m o n s tra tio n , w ith 3“ and 6 - cm, b a n d , o f th e " tu n n e l
e ffe c t" ,
1 .2
The e v a n e s c e n t wave:
th e tim e dependent pro b lem and aim
o f th is in v e s tig a tio n
I t fo llo w e d n a t u r a l l y from th e s e s t u d i e s to pose th e q u e s tio n
"How lo n g does i t ta k e th e e le c tr o m a g n e tic w a v e / p a r t i c l e to t r a v e r s e
th e " c u t - o f f " s e c t i o n / b a r r i e r ? "
I n th e p a r t i c l e c a s e i t was fo u n d
t h a t some im p o rta n c e h ad a lr e a d y b e e n a tta c h e d t o th e q u e s tio n i n
th e stu d y o f t h i n f i lm d e v ic e s b a s e d on th e tu n n e l e f f e c t phenomenon
( 9) >19^ 2) ^ (10, 1965) ^
o f th e e l e c t r o n t r a n s i t tim e s was
n e c e s s a r y i n p ro p o s in g th e p o s s ib l e fre q u e n c y l i m i t a t i o n s o f th e
d e v ic e s .
U nder th e s e c irc u m s ta n c e s , i t was f e l t t h a t an answ er to
th e q u e s tio n i n th e a n a lo g o u s w aveguide case w ould be b o th i n t e r e s t i n g
and u s e f u l .
T hus, th e tim e d ep en d e n t c o n s id e r a tio n o f th e e v a n e s c e n t
waves i n w aveguides fyom th e b a s i s o f th e i n v e s t i g a t i o n d e s c r ib e d
in th is th e s is .
-2 0 1•3
( 1 .3 )
G e n e ra l tim e d ep en d e n t c o n s id e r a tio n s and v e l o c i t i e s
F o r an i n f i n i t e l y lo n g s in e wave t r a i n , a p h a se v e l o c i t y h a s
b e en d e fin e d , w h ich , i n g e n e r a l, i s o f l i t t l e p h y s i c a l s ig n if ic a n c e *
I t i s th e sp e ed ,
v ^ , w ith w hich a p o in t o f c o n s ta n t ph ase moves*
I n th e tim in g c o n s id e r a tio n s o f e le c tr o - m a g n e tic w aves, i t
m ust be p o s s ib l e to say t h a t a r e c o g n is a b le p a r t o f th e wave t r a i n
i s s i t u a t e d a t p o s i t i o n z = a a t tim e t = t ^ and t h a t a t tim e
t = t ^ , t h i s r e c o g n iz a b le p a r t i s a t p o s i t i o n z = b .
In d iffe re n tia tin g
one p a r t o f th e wave t r a i n from th e r e s t , a s ig n a l h a s been c r e a t e d .
The tim e o f t r a v e l o f t h i s s i g n a l o v e r th e d is ta n c e (z ^ - z^) i s
(t^ - t^ ) .
I n th e m aking o f t h i s s i g n a l , o th e r f r e q u e n c ie s a r e
in tr o d u c e d and th e s in g le fre q u e n c y w ave, r e p r e s e n te d by exp]( w ^ t Pz) h a s been r e p la c e d by th e g ro u p o f waves
f(o) - CO )*
exp(jcot - yz)»
dco
v'
—
00
where f (w -00^) i s th e fre q u e n c y d i s t r i b u t i o n f u n c t i o n .
T h is h a s
l e d to a d e f i n i t i o n o f group v e l o c i t y and group t r a v e l tim e f o r
wave p a c k e ts r e p r e s e n te d by
( z ,t) *
I t must be s t r e s s e d t h a t t h i s
i s s t r i c t l y a p p li c a b l e o n ly f o r wave p a c k e ts c o m p risin g a l i m i t e d
(^11^
ra n g e o f f r e q u e n c ie s and f o r n e g l i g i b l e d i s t o r t i o n o f th e s ig n a l^
^.
The m ethod o f s t a t i o n a r y p h ase i s u se d i n th e d e r i v a t i o n o f th e
e x p re s s io n f o r group v e lo c ity *
F o r e ac h p o s i t i o n , z , t h e r e w i l l be
a tim e , t , when th e w a v e ^ f d i f f e r e n t fre q u e n c ie s do n o t i n t e r f e r e
d e s tru c tiv e ly .
T h is w i l l o c c u r when th e phase o f th e e x p o n e n tia l
h a s an extremum and th e waves have n e a r l y th e same p h a s e .
From t h i s
c o n d itio n , th e e x p r e s s io n ( t - z d ^ d c u ) = o i s o b ta in e d and th e
-2 1 -
( 1 .3 )
a v e ra g e sp eed o f tr a n s m is s io n o r group v e l o c i t y i s g iv e n by
v_ = d w /
g
d(3
Iw = w o
i
o
An e n erg y v e l o c i t y i s a ls o d e fin e d a s fo llo v fs : e n e rg y v e l o c i t y = tim e a v erag e P o y n tin g v e c to r
a v e ra g e en erg y s t o r e d / u n i t volum e.
I f no a b s o r p tio n i s p r e s e n t^ th e group v e l o c i t y and e n erg y
v e l o c i t y a re th e same.
However, i t m ust be p o in te d o u t t h a t f o r th e e v a n e sc e n t
Vfave, r e p r e s e n te d by exp^(
w ^ t - a z ) , th e p h ase f u n c t i o n h a s
no 2 dependence and th e sim p le s t a t i o n a r y p h ase m ethod d e s c r ib e d
above w i l l n o t y i e l d a g ro u p t r a v e l tim e .
S p e c if ic tim e d ep en d e n t c o n s id e r a tio n s and th e form o f th e
in v e s tig a tio n .
S in ce th e u s u a l c o n c e p ts o f v e l o c i t i e s and t r a v e l tim e s f o r
p ro p a g a tin g w aves a r e n o t a p p lic a b le f o r t h i s e v a n e sc e n t w ave, i t
i s n e c e s s a r y to malce a more d e t a i l e d i n v e s t i g a t i o n .
T here e x i s t
( 12 )
s e v e ra l th e o r e tic a l in v e s tig a tio n s ^
^ o f th e p ro p a g a tio n o f
s i g n a ls i n w aveguides o r m edia w ith s i m i l a r d i s p e r s i v e p r o p e r t i e s
vfhere th e c a r r i e r fre q u e n c y i s n o t c u t - o f f .
ap p ro ach in v o lv e s tr a n s fo rm te c h n iq u e s .
The t h e o r e t i c a l
T hese w ere exam ined i n
d e t a i l f o r p o s s ib l e a p p l i c a t i o n to th e below c u t - o f f c a s e .
However,
m ost w ere i n a p p l i c a b le s in c e th e y w ere a p p ro x im ate e v a lu a tio n s
u s in g a T a y lo r e x p a n sio n o f th e im a g in a iy ph ase f u n c t i o n .
T here a re some w hich a re g e n e r a l l y a p p lic a b le to c a s e s where
th e c a r r i e r fre q u e n c y o f th e s i g n a l i s above o r below th e c u t’- o f f
freq u en cy #
These i n v e s t i g a t i o n s r e f e r s p e c i f i c a l l y to th e
-2 2 -
( 1 .4 )
p r o p a g a tio n o f a \ m i t - s t e p m odulated c a r r i e r wave i n an i n f i n i t e l y
lo n g , u n te rm in a te d g u id e .
C h a p te r 2 i s d e v o te d to d is c u s s io n o f t h i s i n th e c a se where
th e c a r r i e r fre q u e n c y i s below c u t - o f f .
The tra n s fo rm te c h n iq u e s
u se d a re d e v e lo p e d i n d e t a i l and i t w i l l emerge t h a t t h e r e a re two
ty p e s o f s o l u t io n s f o r th e f i e l d q u a n ti t y a t a g iv e n p o s i t i o n a lo n g
th e g u id e :
f i r s t l y , ap p ro x im ate e v a lu a tio n s v a l i d f o r l a r g e v a lu e s
o f tim e and s e c o n d ly , e x a c t s o l u t io n s v a l i d f o r a l l tim e .
These
s o l u t i o n s w i l l be u se d i n c o n s i d e r a ti o n o f th e P o y n tin g v e c t o r .
T h is w ork i s a tim e d ep en d en t s tu d y o f th e p u r e ly e v a n e s c e n t wave*
I t h a d b een i n te n d e d , when t h i s i n v e s t i g a t i o n was e n v is a g e d , t o
in c lu d e some e x p e rim e n ts .
However, w h ile th e r e s u l t s o f C h a p te r 2
a re i n t e r e s t i n g from a t h e o r e t i c a l p o in t o f view th e y do n o t le n d
th e m se lv e s to p r a c t i c a l i n v e s t i g a t i o n .
To o b se rv e th e f i e l d
q u a n t i t i e s a t a p o i n t a lo n g th e **c u t - o f f ” g u id e , an o u tp u t c o u p lin g
d e v ic e i s in tr o d u c e d a t t h a t p o i n t .
I t s i n t r o d u c t i o n w i l l p ro d u ce a
r e f l e c t e d ?/ave and e n e rg y v / i l l no l o n g e r be t o t a l l y r e f l e c t e d a t th e
f r o n t s u r f a c e o f th e c u t - o f f g u id e - t o t a l r e f l e c t i o n ha©
been f r u s tr a te d !
The w aveguide arran g em en t m ost am enable to
t h e o r e t i c a l and e x p e rim e n ta l i n v e s t i g a t i o n i s th e one i n w hich a
s e c t i o n o f c u t - o f f g u id e i s sandw iched betw een s e c tio n s o f p r o p a g a tin g
g uide ( s e e f i g ,2 ( 1 7
)).
The t h e o r e t i c a l s id e o f t h i s problem i s d is c u s s e d and d e v elo p ed
i n C h a p te r 3*
The p o s s i b i l i t y o f c o n s id e rin g t h i s a s th e tim e
d ependent e le c tr o m a g n e tic an alo g u e o f th e quantum m e c h a n ic a l tu n n e l
e f f e c t w i l l be s t r e s s e d .
C h a p te r k c o n ta in s a d e s c r i p t i o n o f th e
e x p e rim e n ta l s i d e , w hich w i l l be c o n s id e re d a s an e x te n s io n o f
-2 3 ( 7)
("t.4 )
H u p e rt and O tt* s v/ork^ ' on th e tim e in d e p e n d e n t problem #
-2 4 -
( 2 .0 )
CHAPTER 2 .
A STUDY OP THE PROPAG-ATION OP A UNIT STEP
liODULATED CAmiER 19AVÈ m A WAVEGUIDE.
2 .0
In tro d u c tio n
The m a t e r i a l o f t h i s c h a p te r i s l i m i t e d to a stu d y o f th e a p p l i c a t i o n
o f a sudden e le c tr o m a g n e tic d is tu r b a n c e to an u n te rm in a te d w aveguide^ w ith
p ro p a g a tio n c o n s ta n t,
Y , g iv e n by y
2
~
2
-
w
2
.
S in ce a
p lasm a medium a ls o h a s a p r o p a g a tio n c o n s ta n t o f t h i s form , some o f th e
r e s u l t s o f t h i s c h a p te r a re a p p lic a b le to p r o p a g a tio n i n t h i s medium, to o .
The g e n e r a l ap p ro ach to th e s o lv in g o f th e p roblem by th e L a p la c e tra n s fo rm
te c h n iq u e i s p u t fo rw a rd .
I t i s found n e c e s s a iy to p la c e c e r t a i n
r e s t r i c t i o n s on th e form o f th e i n i t i a l e le c tro m a g n e tic d is tu r b a n c e and
a f t e r t h i s , th e u n i t s t e p m o d u lated c a r r i e r wave i s in tr o d u c e d s p e c i f i c a l l y ,
P o r th e c a se when th e c a r r i e r fre q u e n c y i s below th e c u t - o f f fre q u e n c y o f
th e w av eg u id e, s e v e r a l d ev elo p m en ts a re d e s c r ib e d i n d e t a i l , w hich le a d to
a p p ro x im ate and e x a c t s o l u t i o n s , whose ra n g e s o f a p p l i c a t i o n i v i l l be g iv e n .
E x p re s s io n s a re found f o r b o th th e t r a n s v e r s e e l e c t r i c and t r a n s v e r s e m ag n etic
fie ld s .
W ith th e s e two i t i s p o s s ib l e to f i n d th e P o y n tin g v e c t o r .
T h is
c o u ld p ro v e more u s e f u l and I n t e r e s t i n g f o r t h i s c u t- o f f c ase th a n a s tu d y
o f th e d i s p e r s i o n o f th e s i g n a l a lo n e , a s i n th e much i n v e s t i g a t e d c ase
above c u t - o f f .
I t i s hoped t h a t t h i s s h i f t o f em phasis makes some
j u s t i f i c a t i o n f o r th e r e s u l t s o f a t h e o r e t i c a l i n v e s t i g a t i o n w hich c an n o t be
v e r i f i e d e x p e r im e n ta lly .
The e f f e c t o f th e i n t r o d u c t i o n o f a d e t e c t o r i n
th e c u t - o f f g u id e h a s b e en d is c u s s e d i n s e c t i o n ( 1 .1 ) and w i l l be p u rsu e d
f u r t h e r i n C h a p te rs 3 and 4*
2.1
I n t e g r a l tra n s fo r m te c h n iq u e s
An i n t e g r a l tr a n s f o r m a ti o n , T ^ g ( x ) ) , r e p la c e s th e f u n c t i o n g (x ) by
its
im agé" &(s) i n th e "im age" s space* v i z ; -
( 2 ; 1)
T (g (x ))
^
=
g (x ) . K ( x ,s ) . ds
=
G( s )
A s p e c i f i c tr a n s fo rm i s d i s t i n g u i s h e d by th e k e r n e l f u n c t i o n , K ( x , s ) , and
th e l i m i t s o f i n t e g r a t i o n ,
F o r exam ple i f IC (x,s)
b
c a l l e d th e r i g h t h an d ed L ap lac e tr a n s f o r m , L, ,
= CD , th e tr a n s fo rm i s
w h ile f o r K (x , s)
=
exp (-jw x ) and
a
=
=-co , b)
e x p (-sx ), a
=
o,
= +00 ^ i t i s th e
F o u r i e r tr a n s f o r m , s = iw .
The in v e r s e o p e r a t io n i s d e f in e d
by T” '^(& (s))
c a se o f th e in v e r s e L ap lace t r a n s f o r m a tio n ,
g(x)
=
-giTj
I
, i s g iv e n b y , f o r x > o
<J + JOD
Jo- _ j(30
(2.
• G(B) . as
.
.
= g (x ) and i n th e
I n t e g r a l tra n s fo rm s a re u s e f u l i n t h a t th e y may be u se d to red u c e p a r t i a l
d i f f e r e n t i a l e q u a tio n s to o r d i n a i y d i f f e r e n t i a l e q u a tio n s o r a lg e b r a ic
e q u a tio n s , w hich may b e e a s i e r to so lv e th a n th e o r i g i n a l e q u a tio n s .
The
b a s ic t o o l o f t h i s t h e o r e t i c a l i n v e s t i g a t i o n i s th e L a p la c e tra n s fo rm a tio n *
2 ,2
The L ap lac e tra n s fo rm
From e q u a tio n ( 2 , i , l )
th e f u n c t i o n g (x ) i s e x p re s s e d a s a c o n to u r
i n t e g r a l , th e p a th o f i n t e g r a t i o n b e in g th e s t r a i g h t l i n e r e a l s
a s shown i n f i g u r e
( 26 ) ,
=
o^,
G-(s) i s r e g u l a r i n th e r i g h t hand s id e p la n e
(sh a d e d r e g io n ) and th e c h o ic e o f
i s a r b i t r a i y p ro v id e d i t ex ceed s th e
r e a l p a r t o f each p o le o f G-(s) w hich u s u a l ly l i e i n th e l e f t hand s p la n e .
S in ce th e image f u n c t i o n G-(s) i s an a n a l y t i c f u n c t i o n o f com plex v a r i a b l e ,
complex f u n c t i o n th e o ry m ethods may be a p p li e d .
T h is i s a f u r t h e r im p o r ta n t
( 2 , 2)
.26-
im a g in a iy s
/
re a l s
F ig u re
The com plex fre q u e n c y , s , p la n e show ing th e
p a th o f i n te g r a t i o n f o r th e i n t e g r a l ( 2 * 1 .1 ) ,
B
re a l p
F ig u re 4,b.
The oo
(p = s/w ^)
th e c o n to u r o f i n t e g r a t i o n f o r i;
o r t< ^ z / c ) .
. K
( 2 , 2)
p r o p e r ty o f th e L ap la c e t r a n s f o r m a tio n ,
P o r i n s t a n c e , u s in g C au ch y 's
th eo re m , d i f f e r e n t p a th s o f i n t e g r a t i o n i n ( 2. 1 . 1) may be u s e d to e v a lu a te
th e i n t e g r a l and d i f f e r e n t p r o p e r t i e s o f th e f u n c t i o n , g ( x ) , may be fo u n d ,
f o r exam ple, i t s a s y m p to tic b e h a v io u r,
2 ,3
L ap lac e tra n s fo rm te c h n iq u e s a p p l i ed to Maxivell* s e q u a tio n s *
M axi?ell*s e q u a tio n s , f o r th e p r o p a g a tio n o f e le c tro m a g n e tic waves i n
v/aveguide, w hich i s ^ d i s s i p a t i o n l e s s and i n f i n i t e i n th e l o n g i t u d i n a l
f\•
d i r e c t i o n a re a s f o llo w s : -
V
A^
+ |i ( d f L / 9 t )
= o
a -R
„ e (3 L /ô t)
V
= o
\/ *^
V
't t
■*-
=
o
=
o
The tim e dependence can b e tra n s fo rm e d o u t o f th e f i e l d e q u a tio n s by
th e L a p lac e tra n s fo r m .
S in ce \ 7 in v o lv e s sp ace c o -o r d in a te s o n ly and i s
in d e p e n d e n t o f tim e , M a x w e ll's e q u a tio n s become
V A E
+
S1J.H
= pR
( x ^ , Xg, 2 ,
VAH
-
seE
=-€■£
( x ^ ,X g , z
V * E
,
g
=
O
o
->CX)
where E
=
^
^ , e x p ( - s t) ,d t.
=
0
and s i m i l a r l y f o r H a n d " K ,
^
(x ^ , Xgj
The q u a n t i t i e s
^ x^,
and
2 ,t ) ^ _ ^ r e p r e s e n t th e i n i t i a l s p a t i a l d i s t r i b u t i o n o f th e
m ag n etic and e l e c t r i c f i e l d s a t t
The v e c t o r s ,
£
=0
and s i s th e complex fre q u e n c y .
= £ ( x ^ , x ^ , z , t ) and
= H. ( x ^ , x ^ , z , t ) a re
th e in s ta n ta n e o u s e l e c t r i c and m ag n etic f i e l d v e c t o r s .
T hese may be
r e w r i t t e n as
"
Vj, (x-,, X
(z ,t)
n = 1 , 2 , 3.
n
where
“
®n ( ^ 1 '
, 0 XX
jlX
XX
XX a re f u n c t i o n s o f th e i n d i c a t e d v a r i a b l e s o n ly and
- 28-
( 2 . 3)
a re th e com ponents a lo n g th e o c lo r d in a te ax es o f £ a n d H .
■^^n = ^ n
-
^2^ -^n
*^^n " \
“ ®n
^2^ ®n *^2»®)'
T hus,
W ith a f u r t h e r t r a n s f o r m a tio n w ith r e s p e c t to th e l o n g i t u d i n a l
Goî-ordinate, z, and i n t r o d u c t i o n o f th e tra n s fo rm e d h o u n d aiy c o n d itio n s ,
th e system o f t r a n s v e r s e m a g n e tic and t r a n s v e r s e e l e c t r i c waves i s fou n d
f o r p r o p a g a tio n a lo n g th e p o s i t i v e d i r e c t i o n o f th e z a x i s .
T h is h a s
b e e n e x te n s iv e ly t r e a t e d by G e r r i l l o ^ ^ and h i s r e s u l t s a re sum m arized i n
A ppendix A.
I f g e o m e tric f a c t o r s and o t h e r c o n s ta n ts a re o m itte d and o n ly th o s e
p a r t s w hich a re f u n c tio n s o f th e complex f re q u e n c y , s , a re r e t a i n e d , a
p r o to ty p e tra n s fo rm e d f i e l d v e c t o r can be c o n s id e re d o f th e form
F ^ (s)
=
P (s ) . exp ( - z ( s ^
+ w
o)
.............. ( 2 . 3 . 1)
Tfhere P (s ) i s th e L a p la c e tra n s fo rm e d b o undary c o n d itio n i n th e tim e
domain a t z = o.
The form o f th e o t h e r components i n A ppendix A may be
sim p ly re d u c e d to
= F ^ ( s) A
F^
2
w here
= sF ( s ) A
o
K
=
(s^
^
F^
=
sF ^ (s )
F,
=
F (s)lC
o
A
..............( 2 .3 .2 )
+
T here rem ains th e i n v e r s io n from th e s domain
i n t o th e tim e
dom ain, in v o lv in g i n t e g r a l s o f th e form
(
F ^ (s ) , e^^ , ds
o-Q-oOO
S in c e th e e x p r e s s io n (2 ,3 # 1 ) may be u sed to d e s c r ib e th e s - tr a n s f o r m e d
G ,t)
-
2rcj
t r a n s v e r s e component o f a p la n e wave p ro p a g a tin g i n a sem i i n f i n i t e ,
l o s s l e s s plasm a th e m ain e f f o r t w i l l be c o n c e n tr a te d on i t s i n v e r s i o n , b u t
v/here p o s s i b l e , i n v e r s io n o f th e o t h e r s i n ( 2, 3*2) w i l l be in c lu d e d .
-2 9 -
(2«4)
2*4. R e s t r i c t i o n s on th e s tra n s fo rm e d i n i t i a l c o n d itio n s i n th e tim e doiaaln^
C a r r i l l o ^ ^ fo u n d , i n h i s v e r y d e t a i l e d e x a m in a tio n o f t r a n s i e n t
phenomena i n w aveguides t h a t th e v e c t o r s i n th e tim e dom ain,
tra n s fo rm e d from th e
e le c tr o m a g n e tic .
En and Hn
andH^^,
i n th e s domain w ere n o t ,? i n og e n e r a l,7
I t p ro v e d n e c e a s a iy to im pose r e s t r i c t i o n s on th e
tra n s fo rm e d v e c to r s
a t tim e , t
e q u a tio n s i d e n t i c a l l y *
a n a l y t i c a l form o f
= z / c to s a t i s f y M axw ell’ s
T h e se , i n t u r n , l e d to r e s t r i c t i o n s on th e
&
t
(o ,s ) ,
( o , s ) , B ^ (o ,s ) and
( p ,s ) - T h is
re a s o n in g i s g iv e n " b r ie f ly i n A ppendix B,
T h iise ^ re su its, w hich a r e l i s t e d
below , sh o u ld be w e ll n o te d a s th e y a re im p o r ta n t.
I.
I f th e a x i a l component o f th e f i e l d v e c t o r s ,
f o r T,H, f i e l d s ( o r
at
at
g =
0
2 = 0 f o r T .E , f i e l d s ) a re g iv e n r e s p e c t i v e l y
a s i n i t i a l c o n d itio n s , th e s e l a t t e r m ust be such t h a t , as s te n d s to
^
in fin ity ,
^ 3 ( o ,s ) o r i t s e q u iv a le n t A ^ ( o ,s ) f o r T,H, f i e l d s
or
( o , s ) o r i t s e q u iv a le n t B ^ (o ,s ) f o r T .E , f i e l d s
m ust behave a s l/s"^
II,
where Y> 3>M = c o n s ta n t,
I f th e space d e r i v a t i v e s w ith r e s p e c t to z o f th e a x i a l components
o f th e f i e l d v e c to r s
i ^
or
(z )
at z
=
( 2 )
at
2 = 0
0
f o r ToH, f i e l d s
f o r T .E , f i e l d s a re r e s p e c t i v e l y g iv e n
as
i n i t i a l c o n d itio n s t h e i r c o rre s p o n d in g tra n s fo rm s must be such t h a t a s s
te n d s to i n f i n i t y
»
E 3(2 )^ ^ f s ) o r i t s e q u iv a le n t
I
or
( o , s ) : f o r T.H , f i e l d s
I'
^ 3(2 ) ' ( o f s ) o r i t s e q u iv a le n t
m ust behave as R /s^
III,
I
^'3( 2) «
^"or T*Ei f i e l d s
w here Y ^ 2, N = c o n s ta n t,
I f A^^g ^ ( o ,s ) o r B ^ ( o ,s ) a re r e s p e c t i v e l y g iv e n a s i n i t i a l
-3 0 -
(2 .4 )
c o n d itio n s i n th e B: dom ain, i t i s e q u iv a le n t to g iv in g th e tr a n s v e r s e
e l e c t r i c component
z -
o r th e t r a n s v e r s e m ag n e tic component a t
o.
It
was
p o in te d o u t p r e v io u s ly t h a t where p o s s ib l e some o f th e
re s u lts
fo r
th e w aveguide problem would be e x te n d e d to th e p ro p a g a tio n
o f a p la n e wave i n a p la sm a .
B e a rin g i n mind t h a t th e s e a r e t r a n s v e r s e
vfave8 , c o n d itio n s I I and I I I w i l l be s t r e s s e d a t th e expense o f c o n d itio n
I,
2 .5
T h is i s m erely to w id en th e scope o f t h i s i n v e s t i g a t i o n .
T y p ic a l i n i t i a l tim e c o n d itio n s and t h e i r s - tr a n s fo r m s .
I f th e i n i t i a l tim e c o n d itio n i s f ( t )
s in ( u ) ^ t) , t h a t i s , a
o
2
su d d en ly sw itc h e d on s in e w ave, i t s tra n s fo rm i s I ’( s )
=
/
(s +
)
w h ile f ( t )
l(t)
=
=
l(t)
1 ( t ) (1 - cos u>^t)) h a s th e tra n s fo rm
i s th e
u n i t s te p f u n c t i o n ,
l(t)
=
=
1
o
fo r
fo r
s(s^
+
t < o
t > o.
These s tra n s fo rm s comply w ith c o n d itio n I I i n th e above s e c t i o n so i t i s
p o s s ib l e to p ro c e e d w ith th e i n v e r s io n o f th e v a r io u s
( e q u a tio n s
( 2. 5 . 1) and ( 2. 5. 2)} to o b t a i n th e f i e l d com ponents,
2 .6
The u se o f t a b u l a t e d in v e r s io n i n t e g r a l s and th e c o n v o lu tio n th e o re m .
T a b le s o f L ap lac e tra n s fo rm s a r e o f v e ry l i t t l e
use.
Only a few
i n v e r s i o n i n t e g r a l s w ith i r r a t i o n a l e x p o n e n ts a re l i s t e d and th e s e c a n n o t
be u se d d i r e c t l y even i n th e s im p le s t c a s e ,
Knop"^^ and P o in o e lo t^ ^ , w ith a
s u i t a b l e e x p a n sio n o f th e in p u t f u n c t i o n , c o n s is tin g o f th e u n i t s te p
m odulated c a r r i e r wave, have b een a b le to u se an e x i s t i n g tra n s fo rm p a i r .
T h e ir su c c e ss l i e s i n expanding th e in p u t f u n c tio n i n a co n v erg en t s e r i e s
o f B e s s e l f u n c tio n s o f argum ent ( w t ) o f v a r i a b le o r d e r , th u s
0
00
s in (w ^ t)
=
*'^2n + 1
2 ^
( 2 . 6 . 1)
n=o
w here C
xl
=
c o s(2 n
+
=
oosh* (2 n +
1 )0 ,
0
=
a rc cos (oo /w ) ,
l)a ,
a
=
a rc codi .( w ^/w ^),
0 ^ 0
U) ^
O
c
^ ou
-31-
(2 .6 )
The in v e r s e tr a n s f o r m a tio n becomes
*1 I e x p (-z ^ s
T h is i n v e r s io n i n t e g r a l i s t a b u l a t e d
2
2\ ? , \
+ (o^)^/o)
( 17)
oo
/
\ 2n+1
CO
0
and th e f i n a l s o l u t i o n i s
2n+1
f ( z , t ) = 1 ( t_ z /o ) .2 % ( _ l ) "
n=o
"
"^2n+1
* o « e o ( 2 #6« 2 )
Case and H a s k e l l ^ h a v e
s in c e shown t h a t t h i s s e r i e s may be r e w r i t t e n
a s th e sum o f two Lomnel f u n c t i o n s , w hich have b e en t a b u l a t e d f o r
An im p o rta n t theorem r e l a t i n g to th e L ap lac e T ran sfo rm i s th e
c o n v o lu tio n theorem., w hich s t a t e s t h a t
pt
P (s ).G -(s )
=
X(
f(X ) . g ( t - \ ) .d X )
J
q
w here
^*g(t) and f ( t )
a re o f e x p o n e n tia l o r d e r a t t
= 00#
T h is th eo rem w i l l
now be a p p lie d to th e f u n c t i o n P ^ (s) o f e q u a tio n ( 2#3 *1) fo llo w in g a
( 13)
m ethod s u g g e s te d by D oetsch^ ' .
U sin g th e r e l a ti o n ^
JL b ) 2 ) * ( b /a + (a
exp ( ( - a ^ +
/_ ^2 + b. ^2 )«v2
+ b ^ ) ^ ) ^ // (a
•32-
(2 .6 )
v/ 2
exj) ( - a x ) J
(b (x
- 1) ^ ) dx
r e a l v > *- 1
i t can be p ro v ed t h a t
exp ( « z ( s
2
2 i
4 0) ^ ) a)
- exp ( - z s / o )
ZO)
_
( f
i)7 e ) aî
- 1 )^
w here "t =: c t / z
Howeverj th e l a s t r i g h t hand s id e term i s th e L a p la c e tr a n s f o r m , Gr( s ) , o f
th e f u n c t i o n
g ( t ) ss
o
= J ^ (z u )^ (t^ -
fo r o< ^ < 1
) / - ( t ^ - 1)2
f o r t > 1*
The co m plete e x p re s s io n o f (2.3**î) becom es
p
p JL
]? (s)# e x p (~ z (s 4 0) ) ^ /o ) = f ( s ) « exp ( - z s / c )
( a ) ^ z /c ) ,F ( s ) .Z ;( g (t) )
and so ,.f ( z , t ) =
■1
F ( s ) ex p (~ z ( s
4
0)
= jl~ ^ F ( s ) , e x p ( - z s /c ) - (o)^ z/o)X '*'^ ( ? ( s ) G ( s ) )
U sing th e s h i f t theorem ^
' on th e f i r s t r i g h t hand s id e term and th e
c o n v o lu tio n theorem on th e seco n d te rm , th e p r o to ty p e f i e l d v e c t o r becom es
( 2. 6)
- 33«0 ^
;/o
• • 0 0 * 0 0 . • (2o6#3)
where \ i s th e v a r i a b l e o f i n t e g r a t i o n .
(2^ ( 2 1 )
Some a u th o rs^ ^
^ have e v a lu a te d t h i s e x p re s s io n n u m e r ic a lly f o r th e
in p u t f u n c t i o n f ( o , t )
=
l(t)
s i n (w ^t)
a lth o u g h i t i s p o s s ib l e to o b t a i n
th e a n a l y t i c a l s o l u t i o n o f e q u a tio n ( 2 . 6 ,3 ) by th e f o llo w in g m ethod.
U sin g th e s u b s t i t u t i o n s
y =
( X - z / 0) , T
=
( t - z / c ) and Lommel^s
e x p a n sio n
(x +
J . (x + h )2 . x ''/ ^
( " h / 2 )^
m2
-n /2
lir.iirT ,1-n II, ,---i
m=o
th e i n t e g r a l o f e q u a tio n ( 2 , 6 , 3) becom es
T
0)
0
0
z
t i
#
1
(a)^ z /c )“
\ “ o/Y
m =0
W ith th e e x p r e s s io n ( 2 ,6 ,1 ) f o r th e in p u h f u n c t i o n , th e g e n e r a l e x p re s s io n
(2.6)
f o r th e i n t e g r a l becom es
T
-1
#=0,
"^m+1
(Y)-d-Y
Use o f th e t a b i j la te d c o n v o lu tio n i n t e g r a l ,
( 22 )
.a
f o r r e a l i i > o and r e a l v > - 1 ,
and th e s u b s t i t u t i o n m +
1
=
r
g iv e s
Oft
c „ (-i)“ l
2n+1+ r c ) - v , ( i )
r = o
n = o
O O«
Use o f th e s e r i e s
2\
vT
,.(sÜ.-?y. ) / 2 )
(w^) =
r = o
w ith th e s u b s t i t u t i o n s X-a-=; w ^ (t
-s~ s
w ^ (t
a V
0^)2 à i à
« 2/ c) re d u c e s e q u a tio n ( 2. 6 *4 ) to
( 2 »60
- 35-
( 2. 6 )
2n+1
= o
The com plete s o l u t i o n o f ( 2 ,6 .3 ) becomes
= y
-^an+l
A =:
O
S in c e
00
f ( 0 ,t- 2 /o ) ^ /
(-1 )
Cj;^»^2n+1
°))
B =:
T h is i s th e s o l u t i o n c o n ta in e d i n e q u a tio n ( 2 . 6 , 2 ) ,
The e x te n s io n o f th e s e m ethods to th e o t h e r components-,
c o n ta in e d i n
e q u a tio n ( 2, 3 *2) i s r e s t r i c t e d b y th e sm a ll number o f s u i t a b l e t a b u l a t e d
tra n s fo rm s and c o n v o lu tio n i n t e g r a l s .
I t was n e c e s s a r y to make re c o u r s e
to a n o th e r m ethod, w hich w i l l be d e s c r ib e d i n d e t a i l i n th e f o llo w in g
se c tio n s ,
2 ,7
D ir e c t e v a lu a tio n o f th e in v e r s io n i n t e g r a l i n th e com plex fre q u e n c y
p la n e f o r
o< t < z/ c.
Making th e s u b s t i t u t i o n s p
th e i n v e r s io n i n t e g r a l , becom es
=
s / w^,
'(<•=: z w^/o and
X =s
t,
•36 ■
(2.7)
+ jco
CO
o
F(p)«exp(pnr « /c(p +1 ) 2) . dp
2irj
V
cr
Jco
I t oan be shown t h a t f o r x < K ( t h a t i s , t <
f( I ,K)
=
o.
( 2 .7 .1 )
z / g )
T h is oan be done by c lo s in g th e c o n to u r o f th e above
i n t e g r a l w ith a s e m ic ir c le o f i n f i n i t e r a d iu s on th e r i g h t hand s id e o f th e
p a th se e f i g , 4 b ( 2 6 ) , Cauchy* s theorem s t a t e s t h a t i f f(T|) i s a n a l y t i c
i n s i d e and on th e c lo s e d c o n to u r
B
= 0
V
V
JABC
J A
I t can be shown, u s in g one o f th e th eo re m s o f th e i n f i n i t e s e m i- c ir c l e t h a t
rB
0
so t h a t
'-'ABC
= 0
or
f(T ,/< ) = 0
f o r 'z<k
^ A
T h is can be i n t e r p r e t e d a s m eaning t h a t , a t a c r o s s - s e c t i o n , d i s t a n t z from
th e o r i g i n , no p e r t u r b a t i o n a r r i v e s f o r v a lu e s o f th e tim e , t < a / c .
The
i n t e r v a l o ^ t < z / c h a s b e e n c a l l e d th e s i l e n c e zo n e,
2 ,8
D ir e c t e v a lu a tio n o f th e in v e r s io n i n te g r a ,l i n th e com plex fre q u e n c y
p la n e f o r t > a / c ,
F o r v a lu e s o f
t > z /c , th e c o n to u r i s c lo s e d i n th e l e f t h an d p la n e .
H e re , p o le s o f F (p ) may b e p r e s e n t and b e c a u se o f th e r a d i c a l i n th e e x p o n e n t,
t h e r e a re b ra n c h p o i n t s a t p
= + j.
The p p la n e i s a Riemann s u r f a c e
composed o f two s h e e ts c o n n e c te d by a b ra n c h c u t. On t h i s Riemann m a n ifo ld ,
2
JL
th e f u n c t i o n (p
+ 1)% i s a n a l y t i c . I n o r d e r t h a t F (p ) b e a n a l y t i c i n
and on th e c o n to u r, i n d e n t a t i o n s a re made to e x c lu d e th e p o le s o f F (p )
from i t .
F ig u re 5^(37) r e p r e s e n t s a t y p i c a l c o n to u r f o r a tr a n s f o r m whose
( 2, 8 )
b ra n c h c u t
c o n to u r o f
in te g ra tio n
s i n g u l a r i t y o f G-(p)
- R —>00
(T) b ra n c h p o i n ts o f ■jfhi
:u;
T y p ic a l c o n to u r o f i n t e g r a t i o n i n th e p p la n e
> Z / G) .
P a th on f i r s t Riemann s h e e t
P a th on seco n d Riemann
sheet
0
0 —
B ranch p o i n t
-0
B ranch o u t
P o le
s a d d le p o i n ts (p = p )
C ro s s - o v e r p o in ts on th e b ran c h
c u t from f i r s t to se c o n d Riemann
s h e e t.
P ig u re 5b
P a th o f i n t e g r a t i o n th ro u g h th e s a d d le p o in ts
2.8)
•38""
s i n g u l a r i t i e s a re shown,
to th e i n t e g r a l .
c u ts i s z e ro .
\Vhen R->tX), th e s e m i- o ir c le c o n tr i b u te s n o th in g
I n t e g r a t i o n a lo n g c o n n e c tin g c h a n n e ls w ith o u t b ra n c h
T hus, th e i n t e g r a l re d u c e s to i n t e g r a t i o n aro u n d th e
s i n g u l a r i t i e s and a lo n g th e b ra n c h c u t s ,
2*9
S addle p o in t m ethod o f i n t e g r a t i o n , t > z / c , a s y m p to tic s o l u t i o n .
P e a rso n ^ ^ h a s c o n s id e re d d i r e c t e v a lu a tio n o f th e i n v e r s io n i n t e g r a l
i n th e an alo g o u s c ase o f sound waves i n m e ta l tu b e s u s in g a s a d d le p o i n t
m ethod o f i n t e g r a t i o n .
1
2rij
C e rrillo
25
h a s shown how i n t e g r a l s o f th e form
IC p ) . exp ( W( p , p ) ) . dp
Y
w here W(p, i ) = p i
-
and P (p) i s supposed to be f r e e from term s
o f e x p o n e n tia l b e h a v io u r , .may be e v a lu a te d by th e s a d d le p o in t m ethod o r
m ethod o f s t e e p e s t d e s c e n t s ^ ,
E s s e n t i a l l y , th e c o n to u r f o r t h i s
i n t e g r a l i s l i m i t e d to one i n which th e m ain c o n tr i b u t i o n to th e i n t e g r a l
i s from a sm a ll r e g io n i n th e v i c i n i t y o f th e s a d d le p o i n t and th e r e s t
o f th e c o n to u r does n o t c o n tr ib u te s i g n i f i c a n t l y to th e i n t e g r a l .
S ad d le
p o i n t s a re d e fin e d a s th e s o l u t io n s o f dYif / dp = o . The a p p l i c a t i o n
1
IÎ
j
o f th e m ethod r e q u i r e s t h a t |Tf ( p ^ , ! ) | i s s u f f i c i e n t l y l a r g e i n th e
v i c i n i t y o f th e s a d d le p o i n t .
The f u n c t i o n W(p, T ) can be e x p re s s e d
a p p ro x im a te ly as
W (p,T )
= W (P g , T )
+
(P
-
Pg)^ . W ( P g , t
and th e i n t e g r a l re d u c e d to
£ (t)
1
2TTJ
P (p ) . exp(V7*'(p, t ) )
w hich i s f i n a l l y found i n th e form
dp
-3 9 -
(2 .9 )
03
(2irW -Cp^,
T) )=
S in ce i t was r e q u i r e d t h a t
Z
^=O
r(i)
)j
|W
(2 |i) ( (& W "(p .T))t^
B
i s l a r g e , t h i s s e r i e s sh o u ld
converge q u ic k ly and f o r v e ry l a r g e v a lu e s , th e s e r i e s re d u c e s sim ply to
(-r ) »
th e
exp(W ( P g , r ) ) , i C P g ) / ( 2 i t w " ( P g , T ) ) ^
......(2.9.2)
2
JL
th e w aveguide a p p l i c a t i o n , W ( p , r ) = P T K (p
+ i)^
and
2 i.
s a d d le p o i n t s a re g iv e n h y p = jkj / (1 - ( K/ T' ) ) ^
and
IW ( P g f T ) I =
For
,
F or fix e d
Pg
=
+ joo.
^ ( ( : v/
k) ^
-
1^ ^
( t h a t i s , a t a p a r t i c u l a r p o i n t , z ) v/hen T
When
, p^
=
+ j.
=
K,
T h e r e f o r e , th e s a d d le p o in t
c o rre sp o n d in g to th e p o s i t i v e r o o t ahoye moves a lo n g th e im a g in a ry p a x is
from +joo to + j a s th e tim e in c r e a s e d from t
The p a th s i n
th e p p la n e f o r which th e
c o n s ta n t and e q u a l to th e v a lu e a t th e
=
z /c to t
go
and
Im ag in ary p a r t o f W ( p , ' t ) i s
s a d d le p o in t a re i l l u s t r a t e d i n
f i g , 3h ( 3 7 ) *
when
=;
The c o n t r i b u t i o n to th e i n t e g r a l from th e s a d d le p o in ts
/ Q Q
0 /0
F(p)4lp)== (^cu^/o)^)(p + w / a ) ) ( s e e s e c t i o n ( 2 .5 ) ) i s g iv e n hy
( x ,T ) =
exp (jT (1 -K ^ /T ^ )2 ) + jn-/4.) . (1 _ (.K /t)^)
( a r T ^ ( 1 - ( K A ) ^ ) % /K ^ ) ^ ((
f
) ^ (1 -(k /t)^ )-1 )
C
-4 0 "
exp (jTT (1 - ( k / 't ) ^ ) ^ + jir /4 ) •
( 2 ,9 )
- ( k /t)^ )
(S t ^^(1 - ( kA ) ^ ) V ^ ) ^ ( ( co /a > J ^ (1 -(K /T )2 )" 1 )
and s u b s t i t u t i n g f o r
t
and
k
t h i s re d u c e s to
f ^ ( z ^ t ) = (2co^c/rr2 (o^)^ o cos ( W g t ( l - ( z / o t ) ^ ) 2 +
)
" ( ( ° V zJ^-1 ) V ( ( o t / z ) ^( 1 -(coy<0^ )^ )-1 )
OOOOOOOO
(^2o9*3)
w hich i s th e r e s u l t o b ta in e d by H a s k e ll and C ase^^ i n t h e i r c o n s id e r a tio n
o f t r a n s i e n t s ig n a l p ro p a g a tio n i n l o s s l e s s , i s o t r o p i c plasm a m edia and a
pi
s i m i l a r r e s u l t to t h a t o b ta in e d by P e a rso n , I t w i l l be s t r e s s e d a g a in
t h a t t h i s sa d d le p o in t m ethod o f e v a lu a tin g th e in v e r s io n i n t e g r a l i s
n
j
a p p lic a b le o n ly when |¥ ( p^,% ) j i s l a r g e , P o r th e a p p ro x im a tio n to
be v a l i d f o r tim e s n e a r th e w a v e fro n t d e fin e d by
^ (= z
/ o) m ust be s u f f i c i e n t l y la r g e to make
z /c t
=
1 , th e n
| W (p ^/T ) |
la rg e .
On th e o t h e r h a n d , f o r s m a lle r v a lu e s o f /< , th e a p p ro x im a tio n w i l l be
v a l i d f o r c o m p a ra tiv e ly l a r g e v a lu e s o f I , t h a t i s , f o r tim e s much
g r e a t e r th a n t h a t g iv e n by t «
g /c .
The e x te n s io n to th e o t h e r com ponents o f e q u a tio n (2 .3 * 2 ) may be
2
1
a cco m p lish ed by r e p la c in g th e f u n c t i o n P ^(p) = P (p) ,e x p (-K (p
+ 1)^
P ( p ) .e x p
^j^(p) •
=
(*T0Cp) ) 3-n e q u a tio n ( 2 .9 ,1 ) by th e a p p r o p r ia te f u n c t i o n ,
T h is s a d d le p o in t e v a l u a t i o n i s alm o st e n t i r e l y d ep en d en t on th e
b e h a v io u r o f th e e x p o n e n tia l f a c t o r , th e m u ltip ly in g f a c t o r P (p ) a p p e a rin g
i n th e o u tp u t s o l u t i o n a g a in i n th e form o f a m u ltip ly in g f a c t o r , P (p ^) to
{{
JL
th e f a c t o r , exp (W (p^,x)) / ( 2 ttW (p ^ ,X ))%,
S ince th e f u n c t i o n P ^ (s )
a re o b ta in e d from ^ ^ ( s ) by m u ltip ly in g by th e a p p r o p r ia te pow ers o f s and
* 41"
K(=s
2
2 ~
+
(2o9)
th e o u tp u t f u n c t i o n s c o rre sp o n d in g to them a re a ls o
o b ta in e d by m u ltip ly in g th e o u tp u t f u n c tio n o f e q u a tio n ( 2 ,9 .2 ) b y th e
a p p r o p r ia te v a lu e s o f p^ and K (P g) ( p =: q / w ) ,
However, th e s o l u t io n c o n ta in e d i n e q u a tio n ( 2 ,9 .3 ) i s n o t th e
com plete s o l u t i o n to th e p ro b lem s in c e i t w i l l be n o te d t h a t th e p a th
i n th e p p la n e f o r w hich In j ( p i
(p
2
+
1
1)%) = c o n s ta n t = ( v a lu e a t
th e s a d d le p o i n t) a ls o , c u ts th e im a g in a ry a x is a t p o i n ts
fis th e s a d d le p o i n t s ,
and
A^ and A^ g e t n e a r e r to + j .
b e e n c lo s e d when p
and A^,
T h u s,
move i n from i n f i n i t y to w ard s + j , so
T h u s, f o r
+ j
v
= w ^ / w ^ < 1, th e p o le s w i l l
The p o i n ts A^ ^ s.re n o t s a d d le p o i n ts
but^ th e p o i n ts w here th e c o n to u r r e c u ts th e im a g in a ry p a x is and a ls o
c r o s s e s from one Riemann s u r f a c e to a n o th e r , an d p ^^
=
si*
The c o n to u r m ust be in d e n te d now to e x c lu d e th e s e s i n g u l a r i t i e s and
f o r R(p)
2
2
2
2
+ w ^ y/u) Q ) r e s i d u e s o f th e two p o le s make a
c o n tr i b u ti o n
( ^ / 2 j ) . 1 ( t - z (î - co^/cû^ ) ^ / c ) .
0 0
-(ex p (jW g t
(co^ " % ) V o ) - exp
- z
V o ))
(Z.% k-)
T h u s, th e com plete s o l u t i o n i s t h e sum o f th e s o l u t i o n s c o n ta in e d i n
e q u a tio n s (2„.9,3) and ( 2 , 9 , 4 ) .
T h is e v a lu a .tio n o f th e i n v e r s io n i n t e g r a l e x c lu d e s th e o u tp u t
s o l u t i o n f o r tim e s o n ly s l i g h t l y g r e a t e r th a n th e w a v e fro n t a r r i v a l tim e^
2/ 0 o
I n th e n e x t s e c t i o n , t h i s o m is sio n w i l l be rem e d ie d ,
2 ,1 0 ,
E v a lu a tio n o f th e i n v e r s i o n i n t e g r a l f o r sm a ll v a lu e s o f ( t - z / 0) ,.
A m ethod f o r e v a lu a tin g th e in v e r s io n i n t e g r a l f o r th e s e v a lu e s o f
( 28^
tim e , t , h a s b e en p r e s e n te d by Sonm erfeld^ ' and l a t e r u se d by H a s k e ll
( 2T) "
and Case a ls o ^
. I t i s re p ro d u c e d b r i e f l y b elo w .
—42—
(2«10)
I f th e i r r a t i o n a l p a r t o f th e 83{ponent o f th e i n t e g r a l i n
e q u a tio n ( 2 ,7 .1 ) i s expanded i n powers o f ( l / p ^ ) and o n ly te rm s
up to th e f i r s t o r d e r r e t a i n e d , th e i n t e g r a l heoomes
p 0 4- j CA
( Wq /
exp(p(-c-/c) -
27T j )
^ c /2 p ) .P ( p ) .d p
( 2 . 10 , 1 )
u
- d<®
F o r th e u n i t s t e p m o d u lated c a r r i e r w ave, •
2
2
2
2
o (p + W ^ / W ^ ) and assum ing t h a t |pi^> OJ^y«^this
F, (p) =
heoomes F (p )
v / w p^ where
o/ o
v
o
=w / W .
0
0
For th is in p u t, a
h ig h fre q u e n c y e x p a n sio n o f th e e x p re s s io n s f o r th e v a r io u s
components c o n ta in e d i n e q u a tio n s (2 ,3 * 1 ) and (2 ,3 * 2 ) y i e l d s
m u ltip ly in g f a c t o r s f o r th e e x p o n e n tia l w hich a r e f u n c tio n s o f
in v e r s e pow ers o f p ,
Thus th e above i n t e g r a l ( 2 ,1 0 ,1 ) becom es
0 H- .jo0
®3cp(p('t - ^ ) - / c / 2 p ) , d p / 8m
( c o n s t a n t / 2 j -rr )
V
0 -H Jcr>
By making s u i t a b l e s u b s t i t u t i o n s and deform ing th e p a th o f i n te g r a .t i o n
i n th e p p la n e i n t o a c i r c l e o f l a r g e r a d iu s (p w i l l be v e ry l a r g e on
t h i s c i r c l e ) i t i s p o s s ib l e to red u c e t h i s i n t e g r a l to an im p o rta n t
in te g r a l re p r e s e n ta tio n o f a B e sse l fu n c tio n .
.F o r n= o , th e o u tp u t
s o l u t io n i s , f o r sm a ll v a lu e s o f ( t - z / o) ?
1
^
,t)
( 2 z (t - z / o ) / c )
2z( t-z /c )/o )2 )
( 2 , 10 . 2 )
B om m erfel(^^^^raised an i n t e r e s t i n g p o i n t , w hich i s a p p lic a b le to
s o l u t i o n ( 2 ,1 0 ,2 ,) to o .
From t h i s e q u a tio n th e p e r i o d o f th e i n i t i a l
f o r e r u n n e r i s g iv e n a p p ro x im a te ly by th e f i r s t r o o t o f th e B e s s e l
( 2 . 10)
fu n c tio n ^ J ^ ( b ) , w hich i s a p p ro x im a te ly b%-7r.
T hus, th e i n i t i a l
p e r io d , 2 ( t - ^ ) , i s g iv e n by
2
to =
n c
, Tfhich i s in d e p e n d e n t o f th e c a r r i e r fre q u e n c y
w
o
2
o f th e in p u t b u t dependent on th e c u t - o f f fre q u e n c y , w^, and th e
d i s t a n c e , z , t r a v e l l e d i n th e d i s p e r s iv e medium.
2o11
E v a lu a tio n o f th e a p p ro x im ate s o l u t io n s o b ta in e d t h i s f a r
and th e n eed f o r an e x a c t s o l u t io n .
A t t h i s p o i n t , i t i s a d v a n ta g e o u s to enum erate th e s o l u t i o n s
found i n th e p ro c e e d in g s e c tio n s *
z o r c t/z
( t h a t is,% o r
t/ k}
S o lu tio n s f o r l a r g e v a lu e s o f
may be found th ro u g h e q u a tio n (2 .9 * 2 )
and s p e c i f i c a l l y f o r th e u n i t s te p m od u lated c a r r i e r wave i n p u t , th e
o u tp u t,
f ( z , t ) , o f e q u a tio n s (2 .9 * 5 ) and ( 2 .9 .4 ) a r e g iv e n .
E o r v e ry sm a ll v a lu e s o f ( t - z /o ) th e o u tp u t s o l u t i o n f o r th e
same in p u t f u n c t i o n i s g iv e n i n e q u a tio n ( 2 ,1 0 .2 ) *
U n le ss
2 i s v e ry l a r g e , th e ra n g e s o f a p p l i c a t i o n o f th e s e
two o u tp u t s o l u t i o n s w i l l n o t o v e rla p and th e s o l u t i o n i s in c o m p le te .
A s o l u t i o n v a l i d f o r a l l v a lu e s o f z and t i s needed and t h i s i s
fou nd i n su b se q u e n t s e c t i o n s .
The s o l u t io n s w i l l n o t be a s g e n e r a l
a s th e ap p ro x im ate s o l u t i o n s i n t h a t a p a r t i c u l a r p a i r o f com ponents,
w i l l be c o n s id e re d w ith a n in p u t f u n c tio n o f th e u n i t s te p
m o dulated c a r r i e r wave*
2*12
E x act e v a l u a t i o n o f th e i n v e r s io n i n t e g r a l i n th e
fre q u e n c y pla n e f o r v a lu e s o f tim e , t > ^ c .
R ubinow ic2 ^
and W ait and Spies^^^^ have u se d a d i f f e r e n t
c o n to u r i n th e com plex fre q u e n c y p la n e , n am ely , one on w hich th e
e]q)onent o f th e i n v e r s i o n i n t e g r a l rem ain s im a g in a iy .
They have
c o n s id e re d i n d e t a i l th e p r o p a g a tio n o f a u n i t s te p m o d u lated
c a r r i e r wave whose c a r r i e r fre q u e n c y i s above th e c u t - o f f fre q u e n c y
o f th e w aveguide
—Zji},-( 2 oi 2)
and have o b ta in e d an e x a c t s o l u t i o n to th e p ro b lem .
However, th e c o n to u r may be u s e d f o r th e below c u t - o f f c a se and t h i s
i s p r e s e n te d i n t h i s s e c t i o n .
I t i s n e c e s s a r y to d e s c rib e th e in p u t
f u n c tio n s p e c i f i c a l l y a t th e b e g in n in g o f t h i s a n a l y s i s and t h i s i s
f(t) -
l(t)
s in ( w ^ t )
( 2 , 1 2. 1)
w hich h a s th e image f u n c t i o n
i'(p ) =
(v y w g ) /
(p ^ +
wywg
'
( 2 . 1 2. 2)
i n th e com plex fre q u e n c y s p a c e , p .
To be a b le to make P o y n tin g v e c to r c a l c u l a t i o n s l a t e r , a p a i r o f
tr a n s v e r s e com ponents, th e e l e c t r i c and m a g n e tic f i e l d s , m ust be known.
W ith t h i s i n m ind, th e tr a n s fo im e d com ponents
î ’o = P ( p ) .e x p ( - k ( p ^ 4. 1) ^
and
p P ^ (p )/
have b e e n chosen.
. . . . . . . ( 2 . 12. 3)
(p ^ + 1)^
. . . . . . . ( 2 . 12 , 4 )
B r i e f e x a m in a tio n o f th e wave com ponents i n A ppendix
A (fro m C e rrillo ^ ^ ^ ^ ) shows t h a t , i f P (p) i s th e p tra n s fo rm e d (p = s/w ^)
i n i t i a l c o n d itio n i n th e tim e domain a t z = o o f th e
tra n s v e rs e e l e c t r i c
f i e l d ( e q u iv a le n t to A^ ( o , s ) f o r T.H, waves) o r th e
t r a n s v e r s e m a g n e tic
f i e l d ( e q u iv a le n t to
( o ,s ) f o r T .E , waves) th e n , f i r s t l y th e
i n v e r s io n i n t e g r a l c o rre s p o n d in g to P ^ (p ) ( e q u a tio n ( 2, 12, 3) r e p r e s e n t s
th e o u tp u t t r a n s v e r s e e l e c t r i c / m a g n e t i c f i e l d and s e c o n d ly , t h a t
c o rre sp o n d in g to Pg^p) ( e q u a tio n ( 2 , 12, 4) ) r e p r e s e n t s th e o u tp u t
tra n s v e rs e m a g n e tic /e le c tr ic f i e l d .
I t can b e checked e a s i l y t h a t th e
tra n s fo rm e d i n i t i a l c o n d it i o n , P (p ), co m p lies w ith th e r e s t r i c t i o n I I
and I I I o f s e c t i o n ( 2 . 4 ) .
P o r p la n e w aves p ro p a g a tin g i n a plasm a P ^ (p ) may be ta k e n to
r e p r e s e n t th e t r a n s v e r s e m ag n e tic f i e l d and P p (p ) th e tr a n s v e r s e
e l e c t r i c f i e l d ( s i n c e t h e i r r a t i o , th e wave im pedance e q u a ls th e
45'
y /v^
ra tio
where y = (p
(2.12)
+ 1) 2/ 0 ). T h u s, th e r e s u l t s o f t h i s
s e c t i o n w i l l a p p ly to th e p lasm a p r o p a g a tio n .
A t t e n t i o n w i l l he
drawn to t h i s p o in t a t a l a t e r a p p r o p r ia te tim e .
The i n v e r s io n i n t e g r a l f o r P (p ) i s
r{*
O
9
= ( V ^ r r j) ^
e x p ( p T - /< r ( p
i
d p .v
+1)2)
^
(p
+
"Jco
0400000.
Making th e s u b s ti t u t io n p = (
where o ^
W= pt
27r
and
- j cosi|/ f
p
( 2. 12. 5 )
p s in \j/) /( l -
2x4
p )
, th e e x p o n e n t,
- /< (p ^ + 1)2 becom es - j X (1 - g ^ )^ ooslj/ w hich i s alw ays
im a g in a ry .
The p a th o f i n t e g r a t i o n i s i l l u s t r a t e d i n f i g u r e 6(4&)
The i n t e g r a l may b e re d u c e d to
2ir
. ,
zA
.
exp ( - j v ( l - g ) . c o s t ) — —
'( V 2 ir j)
+ Y q ). a*
'........——
. ( 2 . 12 , 6 )
w here dp = d \|/. (g-1 / | ) / 2 ,
and
^ = e x p ( j \ ^ ) ( ( l + |3 ) / (1
Y
V y,
''o
-
j (1 - % )
fo r
< 1 o r CO < co
0
0
0
The i n t e g r a l (g- 1 2 .6 ) may he expanded i n to th e
sum o f two i n t e g r a l , I . and
-2 rr
^■1+^2
"
w here
*
exp(w),d\|f.YpÇ
2-n-j
( g V o)
^ 2'rr
1
e x p (w ).a * .g
'
0
Y o te ^ “ V Y ^)
@ ))
( 2 . 12)
- 4 j6*
im a g in a ry P 4
@
b ra n c h p o i n t
b ra n c h o u t
R
CO
0
p o le o f P (p)
c o n to u r o f
in te g ra tio n
re a l p
0 - j
V
0 _ j
The f o c i o f th e e l l i p s e a re + j
When z / c t
^ , th e e l l i p s e d e g e n e ra te s
i n t o th e s t r a i g h t l i n e b etw een + j
Pigure 6 ,
E l l i p t i c a l c o n to u r o f i n t e g r a t i o n i n th e
com plex p p l a n e , alo n g w hich th e exponent o f
th e i n t e g r a l ( 2 ,1 2 .6 ) i s im a g in a ry
(2.12)
47-
U sing a T a y lo r e x p a n sio n f o r th e f a c t o r s
(
“ 1 /y ' ^ ) I n
i n t e g r a l become
= ( j/2 r r )
( ^
2
-
2 *^i
y ) ” and
Q
I . and I „ s in c e ||( > |y j f o r t
> z / o , th e
2rr
exp(w ), d t . (Y o /5 )(1 + (Y y S ) ^-«-(Yg/S)^ « » • • )
o
I
CO
(j/2ir)
\2lT
ex p (w ).cb |f,(Y y S )
2n+1
n=o
2 IT
Ig
= ( j/2 ir) 2
n=o
S in c e
exp (w)o
(V ^ Y ^ )
2n+1
g = exp ( j + ) . ((1 + / ? ) / (1 - j S ) )
n 2rr
exp
©os ( i) '^ # ( 2 n + 1 ))
h=o
A Sir
exp ( - 0 t ( 1-/3 )^. eos(\}f) - #
n=o
-4 /(Y Y _u )
where
Y
== ((1 -f g ) Z ( 'l'ig ) )
2n+'
(2n+l))-
-ii-6-
( 2 . 12)
U sin g t h e i n t e g r a l r e p r e s e n t a t i o n o f th e B e s s e l f u n c t i o n '
rv 2tt
= (V2it)
exp
cos 0 « jn 0 ) j^odG
^1 + I g =
( Y o /Y j) '" " '
n =:0
1
n=o
00
n= o
A fter su b s titu tio n fo r
^ |3 , Y , th e f i n a l e x p r e ss io n f o r th e
output becomes
CO
f^ (s ,t) = 2 %
( - 1 ) " J 2 n + l ( W c t ( 1 - ( s / o t ) 2 ) 3 ) . C^.
n=o
( ( t - z / o ) / ( t + 2/
0
)
( 2 . 12. 7)
where
0
c o s ( ( 2n + l ) 0 ) and cos 8 = '^o ~
T h is e x p re s s io n i s th e same a s t h a t c o n ta in e d i n ( 2^642) .
\Vhereas
i t was n o t p o s s ib l e to p ro c e e d w ith th e i n v e r s i o n o f o t h e r components
"49 -
( 2 . 12)
i n s e c t i o n (2 ,6 ) i t i s p o s s ib l e h e r e .
The in v e r s io n i n t e g r a l i s
exp (pT-%(p + l ) 2 ) » a p .
fr> (%,?) = ( 1 /2 i r j )
Odd 0 9 0 0
( 2 . 12 . 8 )
U sin g th e p r e v io u s s u b s t i t u t i o n s o f t h i s s e c t i o n ,
P /( p
+l)^=(^f
+ l / ^ )/C 5 —1/ Ê }
&I1&
exp(W)
= - (VSrrj)
U
Y o (p /-Y o )(S ^ -V Y f)
(s + Vs)
(s - 1/s)
d\jf
.( 2 . 1 2 . 9 )
(V w )
V
» (V (^ -Y q )(^ '" V y q ) “’V (^ + y ^ ) (^ + i /Y q ) ) » #
I t can be shown t h a t
-3 0 -
( 2 . 12 )
2
(r-i)(g + i/ç )
k - ,/0 .
0'
(Ç“Yq )(Ç “ V Y o )
and
.2
(g ~1)(S -t-l/£)
co
w
(g -1 /g ) -
(£+Y o )(€ + V Y o )
(g - 1 )
T g ïT lT g T ïT O
o
T hus, th e i n t e g r a l becomes
n2rr
exp(lV). Vg . £ . ( V ( g - Y o ) ( S “l/Y o )+ V (5 -tY o )(S + l/Y o ))
w hich c a n b e r e a r r a n g e d th u s
r\ 2'ir
♦ « (l/2 rrj)
e XI) (W) .V g . (-Y ^ 2 g / ( 1-Yg ) (5^-Y p)
Jo
+ 2 g /(l-ro )(g ^ -l/Y o ))* #
U sin g th e T a y lo r e x p a n sio n f o r t h e f a c t o r s
,2
2s-1
2 s-1
(g - Y .)
and (g - 1/ y „)
a g a in .
2rr
fg (K ,T ) = ( - 1 / 2 rr)
u
(1 - v ! f
ÛO
(
+
(1/Y^
+
-5 1 -
Use o f th e e x p r e s s io n , g
( 2 .1 2 )
= exp( j
) . ( ( l + g ) / ( l - B) ) ^
and th e i n t e g r a l r e p r e s e n t a t i o n o f th e B e s s e l f u n c t i o n
a g a in le a d s to th e fo llo w in g e x p re s s io n f o r f ^ ( /c , t ) ,
y
■
T
w here
CO
^ i r - X
0
h=o
t.
=
w^t^
g =
z / c t , and
y = ( ( 1+ g ) / ( i - g ) ) ^
The f i n a l e x p re s s io n f o r th e o u tp u t becomes
( ...)
= Ù _— — ^
n=o(1-(W g/co^)
. ((t -
'2 . . ,
( ...( 1 - ( ./o t) = ^ ) * ) . V
z /o )/(t +
( 2 .1 2 .1 0 )
Tfhpre
2 .1 3
= s i n ( (2 n + 1 ) 8 )
and oos 0 = v
= Wg/W g.
The com plete s o l u t i o n f o r th e t r a n s v e rs e e l e c t r i c and
magn e t i c f i e l d s .
The o u tp u t s o l u t io n s f o r a p a i r o f tra n s fo rm e d com ponents,
have b e e n fo u n d - e q u a tio n s ( 2 ,1 2 .7 ) and ( 2 . 1 2 , 1 0 ) , However, th e s e
a re n o t th e com plete s o l u t i o n s f o r th e t r a n s v e r s e e l e c t r i c and
m ag n etic f i e l d s s in c e , f o r th e sake o f c o m p actn ess, a l l f a c t o r s n o t
in v o lv in g th e complex fre q u e n c y were o m itte d .
The s o l u t io n s w i l l now
"52-
(2.13)
bw w r i t t e n i n f u l l «
The above a n a l y s i s h a s b e e n c o n fin e d to th e s i t u a t i o n i n w hich
th e t r a n s v e r s e component i s g iv e n a s an i n i t i a l c o n d itio n i n th e tim e
domain a t z = o«
Aif
T h e r e f o r e , u s in g th e e x p re s s io n s on p ag es A2. and
. . c o n ta in e d i n A ppendix A and th e o u tp u t f u n c tio n s
f o ( 2 , t ) and f g ( z , t ) o f s e c t i o n ( 2 , 1 2 ) , th e o u tp u t s o l u t io n s f o r th e
tr a n s v e r s e w l e o t r i c and m ag n e tic f i e l d s a re r e s p e c t i v e l y :
l 'o r ToH, Waves
(Zft)
( 2 ,1 3 .1 )
Ü s Y j.
Ce f g ( g ,t )
n - i l
f ü V ~
ÔXJ L 2
F o r ToE, Waves
C p fg ( z , t )
h^
ax. i\. 2
(2.13)
-5 3 -
tL
(z,t)
F o r p la n e waves i n a p lasm a
%
d
,
^2 ■' " lo '^ 2 (^1^)
The e x a c t s o l u t i o n s , f ^ ( z , t ) and
( z , t ) o f e q u a tio n s (2 .9 * 7 ) und
( 2 . 9 . 0 ) may he u s e d i n e q u a tio n ( 2 , 13. 1)*
For la rg e
k((t//<)
2
3/2
1T » th e r e i s th e a p p ro x im ate o u tp u t
s o l u t io n o f e q u a tio n ( 2. 9 * 3 ) and (2 .9 * 4 ) ^ th u s
2oo^c
“"I
'irzûo
2
0
cos (ü)^t ( i ~ ( z / o t ) ^ ) ^ +
((c t/z )
it/
4)® ( ( o t / z ) ^ - 1 )^
(1 - (w^/o)^) ) « 1)
+ sin (c o ^ t)o ex p ( - 2(0)^ - ü)^)V g)* 1 (t-z (1 « (o ^ /w ^ )^ )
and
lITZ Ct)
( ( o t / z ) (1 - (®n/“ o ) ) - 1)
o
9 —
(^ /i/^rt)
+ OOS ( « ^ t ) .e x p (-z(Wg - w ^ ) y o )
O 0 i
0)^(0^}^)
F o r sm a ll v a lu e s o f ( t « z / o ) , th e h ig h fre q u e n c y a p p ro x im a tio n o f
s e c t i o n (2 ,1 0 ) y i e l d s th e f o llo w in g s o l u t io n ,
(2,t) =
(t-z /o )
-5 4 -
and i t may be shown t h a t
’
fg (z^t) =
(2.13)
(z,t).
T hus, s in c e th e r a t i o o f a p a i r o f t r a n s v e r s e f i e l d com ponents o f
T .E , Yfaves, f o r exam ple, i s
=
^10 f g ( z , t ) / f Q ( z , t )
f o r sm a ll v a lu e s o f ( t - z / c ) t h i s r a t i o i s e q u a l to
JL
{ ^ / c )^ .
T h is i s one o f th e c o n d itio n s f o r th e e le c tr o m a g n e tic n a tu r e o f th e
f i e l d com ponents fo u n d by C e r rillo ^ ^ ^ ^ ( s e e A ppendix B ),
a p p l i c a t i o n o f t h i s c o n d itio n i n th e tim e dom ain
The
l e d to th e
c o n d itio n s to be im posed on th e s tra n s fo rm e d i n i t i a l tim e c o n d itio n
i n S e c tio n ( 2 * i ) ,
2 .1 4
N um erical c o m p u tatio n s o f th e e x a c t and ap p ro x im ate f u n c tio n s
I n th e p re c e e d in g s e c t i o n s o f t h i s C h a p te r, th e a p p l i c a t i o n o f
a u n i t s te p m o d u lated c a r r i e r w are a t z = o i n a w av eg u id e, c u t - o f f
w ith r e s p e c t to th e c a r r i e r fr e q u e n c y , h as b e e n c o n s id e re d i n d e t a i l .
E x act and ap p ro x im ate e x p r e s s io n s f o r th e t r a n s v e r s e e l e c t r i o and
m agnetic f i e l d s a s a f u n c t i o n o f tim e , t , and d i s t a n c e , z , a lo n g
th e g u id e have b e en p r e s e n t e d i n S e c tio n ( 2 , 1 3 ) *
I t w i l l be s t r e s s e d
b e fo r e any f u r t h e r d i s c u s s i o n t h a t f ^ ( z = o , t ) r e p r e s e n ts th e
a p p lie d s i g n a l o r i n i t i a l c o n d itio n o f one k in d o f t r a n s v e r s e f i e l d
component ( t h a t i s , e l e c t r i c o r m ag n e tic ) i n th e tim e dom ain w h ile
f ^ ( z , t ) r e p r e s e n t s th e su b s e q u e n t form o f t h a t component and f g ( z , t )
th e su b s e q u e n t form o f th e o t h e r f i e l d q u a n ti t y ( t h a t i s , m ag n e tic
o r e l e c t r i o ) a s f u n c tio n s o f th e d i s t a n c e , z , and tim e , t .
Exam ples o f th e s e e x p r e s s io n s have b e e n com puted n u m e r ic a lly
u s in g th e E l l i o t t 303 com puter ( U n iv e r s ity o f S u r r e y ) .
I t i s th e
-5 5 -
(2 .1 4 )
aim o f t h i s and su b seq u e n t s e c t i o n s to p r e s e n t g r a p h ic a l i l l u s t r a t i o n s
o f th e f i e l d com ponents r e p r e s e n te d by f ^ ( z ^ t ) and f g ( z ^ t ) a s
f u n c tio n s o f th e n o rm a liz e d d is ta n c e ( z / \ ) o r n o rm a liz e d tim e
W^t f o r v a lu e s o f th e p a ra m e te r f ^ / f ^ .
Em phasis w i l l be p la c e d
on th e e q u iv a le n c e o f th e e x a c t e x p re s s io n s f ^ ( z , t ) and f ^ ( z , t ) and
th e ap p ro x im ate ones f ^
(c t/z ).
( z , t ) and f ^ ( z , t ) f o r l a r g e v a lu e s o f
The re p la c e m e n t o f th e i n f i n i t e sums o f B e s s e l f u n c tio n s
by th e sum o f two tr ig o n o m e tr ic f u n c tio n s sh o u ld be an a i d to
d is c u s s io n o f th e s e t t i n g up o f th e s te a d y s t a t e s i g n a l s .
I n o r d e r to keep th e n a r r a t i o n c l e a r and c o n c is e , a p a r t i c u l a r
p a i r o f t r a n s v e r s e com ponents w i l l be c h o sen .
The
mode w i l l be
c o n s id e re d i n p a r t i c u l a r and th u s i n e q u a tio n ( 2 .1 3 .1 ) f o r T .E .
w aves,
= 1, o = c
[ iç
=
(f)
(Y )
and
w here b i s th e t r a n s v e r s e b ro a d d im en sio n and k = y r / b ,
/
2
\
°
f ^ (z ,t) = t - i y / n l / k ^ ) .
/ b ) . s i n ( tt y /b ) j . . . . . , . . . . ( 2 . 1 4 . 1 )
and
f g ( 2 ,t) =
6 ^ / ^ ( l / k ^ ) . ( 7 r / b ] . sin (iT 'y /b )
. . . . . . . . ( 2 .1 4 .2 )
F o r p la n e waves i n a p lasm a ,
and
=£
. . . . . . . . . . ( 2 . 1 4 .3 )
However, to k e e p a c e r t a i n amount o f g e n e r a l i t y , t h e q u a n t i t i e s on
th e l e f t hand s id e o f e q u a tio n s ( 2 . 14. I ) ^ ( 2 . 14.2 ) and ( 2 .1 4 .3 ) w i l l
be r e f e r r e d to a s th e t r a n s v e r s e m agnetic ( f ^ ( z , t ) ) and e l e c t r i c
( f g ( z , t ) ) f i e l d com ponents r e s p e c t i v e l y .
I t w i l l b e u n d e rs to o d t h a t
th e com plete e x p r e s s io n s o f t h e s e e q u a tio n s g iv e th e components
lit and i f o r th e w aveguide and th e p lasm a,
y
X
•-
(2 ,1 Jf)
The f u l l cu rv e i n f i g u r e 7 (5 ? ) i s a g rap h o f th e tr a n s v e r s e
m agnetic f i e l d ( f ^ ( z , t ) ) v e rs u s th e n o rm a lis e d tim e , w ^ t , f o r th e
p o s i t i o n i n th e w aveguide o r p la sm a , s / X ^ = ,3 1 ? o r oj^s/c^ = 1 and
b ro k e n cu rv e v /ith lo n g d a sh e s shows th e v a r i a t i o n
o f th e a p p ro x im a te f u n c t i o n , f ( s , t )
? /ith tim e f o r th e same
v a lu e s o f th e p a ra m e te rs ( s / X^) and ( f ^ / f ) .
T h is cu rv e h a s
b een c o n tin u e d u n t i l i t i s v i r t u a l l y i n d i s t i n g u i s h a b l e from th e
curve r e p r e s e n ti n g th e e x a c t f u n c tio n .
fo r
^
Thus i t can be seen t h a t
6 , th e e x a c t f u n c t i o n can be r e p r e s e n te d by th e
ap p ro x im ate o n e.
C om putations show t h a t f o r 6 ,3 < W ^ t < 9*3 th e
e r r o r i s n o t g r e a t e r th a n 5 i n th e t h i r d d e c im a l p l a c e ,
]?or l a r g e r
tim e th e e r r o r i s s m a lle r .
23 , th e
F o r exam ple, f o r 2 2 < w ^ t <
a p p ro x im ate f u n c t i o n can r e p r e s e n t th e e x a c t f u n c tio n to th e t h i r d
decim al p l a c e .
However, f o r
w ^t ^
10 th e s i g n a l h a s a lm o st
re a c h e d s te a d y s t a t e , s i n (w ^ t),e x p (-C C z ), w hich i s i n d i c a t e d by
th e b ro k e n cu rv e w ith sm a ll d a s h e s .
The d e v ia ti o n from s te a d y
s t a t e i s m ost a p p a re n t i n th e r e g io n o f th e p e a k s.
Thus f o r t h i s
f i e l d com ponent, th e a p p ro x im ate e x p re s s io n becomes v a l i d and th e
s te a d y s t a t e s e t s i n f o r th e same o r d e r o f tim e a p p ro x im a te ly .
However, f o r th e e l e c t r i c f i e l d component t h i s i s n o t so a s shown
i n f i g u r e 8 (3 8 ) where th e f u l l curve i s a g ra p h o f th e e x a c t
s o l u t i o n v e rs u s tim e ,
(^ ^t ^
W ^ t.
C om putations show t h a t f o r
7 th e ap p ro x im ate and e x a c t f u n c tio n s a re i n d i s t i n g u i s h a b l e .
The b ro k e n cu rv e r e p r e s e n t s th e ste a d y s t a t e component
cos(uj^t) ,e x p ( ~ a z) ( f ^ / f ^ ) y / ( l - f ^ / f ^ ) ^ .
I t i s a p p a re n t from a
com parison o f th e f u l l and b ro k e n c u rv e s t h a t i t v f ill be some
tim e b e fo r e s te a d y s t a t e i s reached*
D is c u s s io n o f t h i s b e h a v io u r
(2.14)
-57F ig u re 7 .
.3
The G ra p h o f t h e t r a n s v e r s e m a g n e t i c f i e l d
f o r th e H
m ode.
com ponent v e r s u s n o rm a liz e d tim e ,
25
th e e x a c t fu n c tio n , f ( z , t ) , ^
th e a p p ro x im a te f u n c t i o n , f
( z ,t)
th e ste a d y s t a t e ,
s in
(w t ^ .
ex
.2
15
.1
.0 5
a m p litu d e
0
.0 5
.1
15
2
6
4
F ig u re 7
8
10
12
14
n o rm a liz e d tim e ,
16
18
20
22
24
26
28
-58F ig u re 8 »
_
(2.14)
T he G ra p h o f th e t r a n s v e r s e e l e c t r i c f i e l d c o m p o n e n t v e r s u s D o rm a liz e d t im e ,
z /\= .3 1 7 ,
th e e x a c t fu n c tio n , fg
( z , t ) ------------- t h e
, f o r th e
■ m ode.
= 1.
ste a d y s t a t e ,
cos ( c o ^ t) .e x p ( - a z ) . ( f ^ / f ^ ) / ( l
-
.3
.2
.1
a m p litu d e
a m p litu d e
0
-.1
—*2
-.3
-.4
F ig u re 8 .
n o rm a liz e d tim e , W t
-5 9 -
(2 .1 4 )
w i l l be d e la y e d u n t i l a l a t e r s e c tio n *
I n th e d e r i v a ti o n o f th e s e approxim ate e x p re s s io n s i n
s e c tio n ( 2 * 9 ) ^ ^ q u a n t i t a t i v e c r i t e r i o n f o r t h e i r range o f
v a l i d i t y was n o t deduced.
The o n ly re q u ire m e n t was t h a t th e
f u n c tio n \7” ( s ^ , t ) = ^ ( { ' \ / ^ ) ^ - lj
was v e ry l a r g e and th e
T a y lo r e x p a n sio n o f th e e x p o n en t c o u ld be te r m in a te d a t th e
second te rm .
T h is e n su re d t h a t th e m ajo r c o n tr i b u ti o n to th e
i n t e g r a l came from th e v i c i n i t y o f th e s a d d le p o i n t ,
A d i f f e r e n c e i n th e ra n g e s o f v a l i d i t y o f f ^ ( z ^ t) and
fg
( z/ X
( z , t ) h a s been n o te d ,
q)
F o r a g iv e n r a t i o ( f ^ / f ^ )
and
i t i s p o s s ib l e to approxim ate th e f u n c tio n f o r th e
m agnetic f i e l d by f^ ( z ^ t) f o r s m a lle r v a lu e s o f W t
e l e c t r i o f i e l d can be a p p ro x im a te d by f ^
th a n th e
(z ^ t).
F o r exam ple, i n th e r a n g e , 1 2 .5 < w ^t < '16 ,3 , th e r e i s a
maximum e r r o r o f 1 i n th e t h i r d decim al p la c e f o r f ^ ( z , t ) w h i l s t
th e r e i s a maximum e r r o r o f 3 i u th e t h i r d d e cim a l p la c e f o r
fp ( z ,t) .
T h is i s a ls o t r u e f o r t h i s p a i r o f components f o r a
d i f f e r e n t v a lu e o f th e p a ra m e te r ,
= .5^ z / K ^ = •O317 (
/0
I n th e ran g e 10*8 < w ^ t
<
f^ (z ,t)
in p a rtic u la r fo r f ^ f ^
= *1) p r e s e n te d i n f i g u r e 9 ( 6 0 ) ,
1 4 ,2 3 ,
f^
(z ,t)
can r e p r e s e n t
c o r r e c t l y to th e t h i r d d ecim al p la c e w h ile
( z ,t ) has
a maximum e r r o r o f 4 i u th e t h i r d d ecim al p la c e *
T hus, i t may be c o n c lu d e d t h a t th e a p p ro x im ate e x p re s s io n s
can r e p r e s e n t th e e x a c t f u n c t i o n s f o r th e m ag n e tic and e l e c t r i c
fie ld s .
The use f u l l n e s s o f t h i s e q u iv a le n c e w i l l be em p h asized
in a l a t e r s e c tio n .
—
60—
F ig u re 9 »
12,
T he G-rg-nh o f t h e t r a n s v e r s e n a g n e t i c a n d e l e c t i v e f i e l d
m ode.
^ o / ^ c = -5
= -0 3 1 7
c o m p o n e n ts v e r s u s n o r m a l i z e d t i m e ,
W .z /c o
, f o r th e
=
m a g n e t ic f i e l d
c o m p o n e n t, f ^ ( z , t ) ,
e le c tric f ie ld
c o m p o n e n t, f _ ( z , t )
a m p litu d e
a m p l it u d e
Î
i
F ig u r e 9
n o rm a liz e d tim e , W t
.0
—6 i *"
( 2 e 1 ^4')
F i n a l l y , an e v a lu a tio n o f th e a p p ro x im a tio n o f th e f u n c tio n s
f o r sm a ll v a lu e s o f ( o t / z - 1 ) , t h a t i s , j u s t a f t e r th e a r r i v a l
o f th e w a v e fro n t, w i l l he made.
F o r th e sm a ll d i s t a n c e , z ,
ch o sen , w hich w ere f r a c t i o n s o f c u t - o f f w a v e le n g th s , i t was
found t h a t th e ap p ro x im ate f u n c tio n s c o u ld r e p r e s e n t th e e x a c t
f u n c tio n s w ith l e s s th a n 10^ e r r o r f o r v a lu e s •
1 < ( c t / z ) < 1 ,l 6 , S in c e ( c t / z ) = (w t /
o
^ o' '
O '
(w z / c ) )
O d
th e n f o r
f ^ / f ^ = .5 and lu z / c^ = 1 f o r i n s t a n c e , th e range o f v a l i d i t y
becomes 1 < ii^t <
1 .1 6 .
S in ce i n f i g u r e s 7 and 8 a ra n g e o f
o < w ^ t < 28 h a s b een c o n s id e r e d , i t i s a p p a re n t t h a t th e
a p p ro x im a tio n i s v e iy l i m i t e d i n i t s ran g e o f a p p l i c a t i o n and no
f u r t h e r d i s c u s s i o n v d .ll b e m ade.
2 .1 5
V a r i a t i o n o f th e t r a n s v e r s e f i e l d com ponents w ith tim e
(oio’t) ^ f o r g iv e n v a lu e s o f ( f ^ / f ^ ) and d is ta n c e ( z / X ^ ) .
The t r a n s v e r s e m a g n e tic f i e l d , vfhich i s th e i n i t i a l
c o n d itio n i n th e tim e dom ain a t z = o , i s g iv e n by 1 ( t ) . s i n
.( W ^ t ) .
From t h i s a re d e riv e d th e o t h e r com ponents.
I t i s th e
p u rp o se o f t h i s s e c t i o n to s tu d y th e tr a n s v e r s e m ag n etic and
e l e c t r i c f i e l d s a s f u n c tio n s o f th e n o rm a liz e d tim e . W t . f o r
^ o '
g iv e n v a lu e s o f ( f ^ / f ^ ) and (z/X ^ ) and, i n p a r t i c u l a r to f i h d when
i t i s p o s s ib l e to re c o g n iz e th e ste a d y s t a t e s i g n a l .
From th e
p o in t o f view o f p r e s e n t a t i o n , i t has p ro v e d re a s o n a b le t o p l o t
th e f i e l d com ponents i n th e ran g e o < W ^t < 28.
th e p a ra m e te rs
Thus v a lu e s o f
and ( z / X ) have b e e n chosen so t h a t e i t h e r
th e f i e l d com ponents show a te n d e n c y to th e s te a d y s t a t e i n t h i s
ra n g e , o r f a i l i n g t h i s , t h a t th e y can be r e p r e s e n te d by th e
ap p ro x im ate e x p re s s io n s by th e tim e W t
^
28,
( 2 . 15 )
-^ 2 -
A n a t u r a l s t a r t i n g p o i n t f o r d is c u s s io n i s th e form o f th e
two com ponents a t th e p o i n t z = Oo
The im p re ss e d component i s
th e t r a n s v e r s e m a g n e tic f i e l d i n th e form o f th e u n i t s te p
m o d u lated c a r r i e r wave.
The t r a n s v e r s e e l e c t r i c f i e l d vfhioh
d e riv e s from t h i s , i s p l o t t e d a s a f u n c tio n o f th e n o rm a liz e d
tim e f o r o <
w ^ t < 50 i n f i g u r e 10(63 ) , w ith f ^ / f
th is c u t-o ff
g u id e a t z =o, th e
= ,5 ,
For
s te a d y s t a t e t r a n s v e r s e e l e c t r i c
f i e l d i s g iv e n by
cos (cu^t) . ( f ^ / ^0
0
( f ^ / f^ ) ^ ^ ^
and i s i n d i c a t e d by ^
th e b ro k e n curve i n f i g u r e 1 0 :w h i l s t i t can be se e n t h a t th e r e
i s a s tr o n g te n d e n c y to th e s te a d y s t a t e , i t i s a p p a re n t t l m t ,
even a f t e r a p p ro x im a te ly 8 o s c i l l a t i o n s o f th e c a r r i e r w ave,
th e r e i s some d e v ia tio n from s te a d y s t a t e , p a r t i c u l a r l y i n th e
v i c i n i t y o f th e pealcs.
I t i s e x p e c te d t h a t f o r z > o , a s th e s ig n a ls become d eg rad ed
by d i s p e r s i o n , so t h i s d e v ia tio n from th e s te a d y s t a t e w i l l become
more m arked.
F o r ( z / X ) - ,3 1 7
' '
O
( w z /c
^ 0
= I ) and f / f = ,5 ,
0
' 0-^0
^
th e m ag n etic f i e l d component h a s re a c h e d th e s te a d y s t a t e i n
th e ra n g e o f w ^t p l o t t e d i n f i g u r e 7(5 7 ) , w h ile th e e l e c t r i c
f i e l d component i n f i g u r e 8(58 ) h a s l i t t l e
s ta te .
te n d e n c y to s te a d y
The o r i g i n o f t h i s b eheiviour can be t r a c e d i n th e ap p ro x im ate
f u n c t i o n s w hich can be u se d to r e p r e s e n t th e f i e l d com ponents
w ith s u f f i c i e n t a c c u ra c y f o r
w ^ t > 10.
e x p r e s s io n s w i l l be r e p e a t e d h e r e .
F o r c o n v en ien ce th e s e
F ig u r e 1 0 .
T he g r a p h o f t h e t r a n s v e r s e e l e c t r i c f i e l d
fo /fc =
-5
-6 3 com ponent v e r s u s n o rm a liz e d ti m e ,
(2 .1 5 )
f o r th e
m ode,
=
th e e x a c t f u n c tio n , f g ( z , t ) .
th e s te a d y s t a t e ,
c o s (ü) t ) . ( f
/ f ) /
(l
- (f
/f
) )
8
.6
•4
4
2
.2
2
-.2
4
- .4
6
-.6
a n p litu d e
0
2
4
6
8
10
F i g u r e 10
20
30
n o r m a l i z e d t i m e , cl> t
'
o
40
50
-6 4 --
(2.15)
+ s i n (co t) o exp (-z(co - co ) ^ /o )o l ( t - z ( l - ( f / f ) )^)
0
O
0
0
0 0 '^'^
= a ( t ) oos (co t 4- 'rr/4) + b
fo r t > z (i -
(f / f
—— —
=
)
2\4-
ana
s i n (co t ) .
(o t / z ) »
and
^
oos(co t)o e x p (~ z (c o ^ -^ ^ )^ /c )"l ( t - z ( l - ^ ( f / f ) ^ ) ^ )
' o '
( t ) oos (co^t 4 %/ih)
^ o
/
o/
^
^
^ o" c/
/ /
+ bg cos (co^t).
B oth a re made up o f th e sum o f th e ste a d y s t a t e te rm ( th e seoond
term on th e r i g h t h an d s id e ) and a term w hich r e p r e s e n t s th e
t r a n s i e n t n a tu r e o f th e s i g n a l o r th e e f f e c t o f th e sudden
STfitohing o n ,
f o r th e m a g n e tic f i e l d ( f ^ ( z , t ) )
t r a n s i e n t term i s p r o p o r t io n a l t o ( c t / z )
t h i s so c a l l e d
f o r v e iy l a r g e
( o t / z ) and th u s f a l l s o f f r a p i d l y v fith i n c r e a s in g tim e .
Thus f o r
W ^t > 20^ t h i s term h a s vexy sm a ll a m p litu d e and i s a n e g l i g i b l e
p e r t u r b a t i o n on th e s te a d y s t a t e te rm (se e f i g u r e 7 ) .
F o r th e
tr a n s v e r s e e l e c t r i c f i e l d (fg (z ^ t))^ th e t r a n s i e n t term i s p r o p o r t io n a l
«Jto ( c t / z ) ” ^ f o r vexy la r g e ( c t / z ) .
T hus, i t d e c r e a s e s much l e s s
quiolcly th a n th e c o rre sp o n d in g te rm f o r th e m a g n e tic f i e l d and
w i l l have a g r e a t e r p e r tu r b in g e f f e c t on th e s te a d y s t a t e te r ^ #
f
T h is i s a rough e x p la n a tio n s in c e th e ten d e n cy to s te a d y s t a t e
-6 5 -
(2.15)
w i l l depend n o t o n ly on th e f a l l i n g o f f o f th e t r a n s i e n t term
b u t i t s a m p litu d e and p h a se compared w ith th e s te a d y s t a t e .
The b e a t in g o f th e t r a n s i e n t and s te a d y s t a t e term s i s a
c o n v e n ie n t way o f e x p la in in g th e p e r t u r b a t i o n o f th e s ig n a l from
th e s te a d y s t a t e »
T h e re fo re th e approDcimate f u n c t i o n s have b e e n
r e - a r r a n g e d to g iv e th e f o llo w in g e x p re s s io n s f o r ( 2/ ÿ )
t > z (i
1 and
-
*
where
(z,t)
=
N ^sin(w ^t
( z ,t)
=
oos ( CO t
and
a r e th e a m p litu d e s and y )^ (t) and cPg(t) a re th e
d e v ia tio n s o f th e
+
kp^(t>)
+ ip ^ ( t) )
p h a se o f th e s i g n a l from th e s te a d y s t a t e
Np =
s in ( (
Ng = (a g
+ bg
= t a n " ''
(a ^
- W ^ ) t +7t A ) ) ®
+ 2agbg cos ( ( u ) ^ - W ^)t +';t / 4 ) ) ^
o o s((
- W ^ )t + ' ? r A ) / ( b ^
-
s in ((w ^ -W ^ )t
+
T rA ))
y) 2 = t a i T
( -
s i n ( ( W ^ -W ) t ^ i r / l i ) / (b ^ +
cos
( ( u > ^ - ix y t + f r / 4 ) )
As th e q u a n t i t é s a^ and a^ te n d to z e ro f o r l a r g e tim e so
and
to z e r o ,
te n d to th e s te a d y s t a t e a m p litu d e and
^ and ipg te n d
F ig u re 1 1 (6 6 ) i s a g ra p h o f th e a m p litu d e
v e rs u s
n o rm a liz e d tim e w ith ( f / f ) = ,5 and
o
c'
z / X ^ = *0317 and .3 1 7 (w ^ z/c^ = .1 and I ) f o r 30< W ^ t < 100,
The a m p litu d e s
b^ and b ^ ,
and
o s c i l l a t e a b o u t th e s te a d y s t a t e a m p litu d e s ,
h a s n o t b e en in c lu d e d on f i g u r e 11 s in c e th e s te a d y
s t a t e fox* th e m a g n e tic f i e l d had b e e n e s t a b l i s h e d i n W^t < 30.
For z / K ^
#317, bg =: ,1 0 3 and f o r z / X ^ w ,0317^ bg = ,4 8 6 ,
-6 6 -
F ig u re 1 1 ,
The g ra p h
o f th e
f ^
,5
/ =
(2 .1 5 )
a ia p litu d e f u n c tio n . N q . v e r s u s n o m a li z e d tim e ,
z/A^ = .317
z /X
=
( w ^ z /c ^ =
,0317 ( w z / c
=
1)
c u rv e
T
.1)
c u rv e
A.
, f o r th e
m ode,
6
5
3
2
N,
2
1
30
40
50
60
n o r m a l i z e d , t i m e , co
F i g u r e 11
70
o
t
80
90
100
-6 7 -
(2 .1 5 )
? o r th e e l e c t r i o f i e l d com ponent, th e a m p litu d e o f o s c i l l a t i o n o f
V.^ a b o u t th e s te a d y s t a t e a m p litu d e a s a p e rc e n ta g e o f th e s te a d y
s t a t e a m p litu d e f o r
w ^t ^
93 i s a p p ro x im a te ly 16^ f o r z /X ^ =
.03") 7 and 77^ f o r z / X ^ = •3 ’)?«
T hus, th e s i g n a l w hich h a s
t r a v e l l e d th e s h o r t e r d is ta n c e seems to b e n e a r e r th e s te a d y s t a t e .
T h is i s b o rn e o u t by th e v a r i a t i o n o f ^ g ( t ) w ith n o rm a liz e d tim e ,
W ^t, f o r th e same p a ra m e te rs (o j^ z /o ^ and ( f ^ / f ^ ) , i l l u s t r a t e d i n
fig u re 1 2 ( 6 8 ) ,
A g a in , th e d e v ia tio n from o r th e o s c i l l a t i o n
a b o u t th e ste a d y s t a t e v a lu e , z ero i n t h i s c a s e , i s g r e a t e r th e
f u r t h e r th e s ig n a l h a s t r a v e l l e d i n th e d i s p e r s i v e medium.
It
can be c o n clu d ed t h a t th e f u r t h e r th e s i g n a l h a s t r a v e l l e d i n th e
medium, th e lo n g e r i t t a k e s to re a c h th e s te a d y s t a t e v a lu e .
T h u s,
th e ap p ro x im ate f u n c t i o n s have p ro v ed v a lu a b le i n p r o v id in g a
method f o r com paring th e s i g n a l w ith th e s te a d y s t a t e o n e .
U n f o r tu n a te ly , th e ap p ro x im ate e x p r e s s io n i s n o t v a l i d f o r z/X ^ = o .
T h is l i m i t s th e u s e f u l n e s s o f th e m ethod.
I t would a p p e a r from a l i t e r a t u r e
su rv e y t h a t c u rv e s f o r a
p a i r o f t r a n s v e r s e com ponents f o r ( f ^ / f ^ ) < ' 1
p u b lis h e d , E ubinow icz ^ a n d
have n o t b e e n
C e r rillo ^ ^ ^ ^ have p u b lis h e d
e x p r e s s io n s f o r a p a i r o f t r a n s v e r s e com ponents f o r ( f ^ / f ^ ) > 1
when a component i n th e form o f a u n i t s te p m o d u lated c a r r i e r
wave i s s p e c i f i e d a.s th e i n i t i a l c o n d itio n i n th e tim e dom ain,
Knop^^^^ and Case^^^^ have p u b lis h e d c u rv e s f o r one component
o n ly w ith th e i n i t i a l form
2 ,1 6
s in ( u ) ^ t ) . l ( t ) f o r ( f ^ / f ^ ) < 1,
The I n s ta n ta n e o u s P o y n tin g v e c t o r
W ith a know ledge o f th e two t r a n s v e r s e f i e l d com ponents, i t
i s p o s s ib l e to p ro c e e d f u r t h e r w ith a c a l c u l a t i o n o f th e P o y n tin g
^
-6 8 -
P iR u re 1 2 .
T h e G -ra c h o f t h e p h a s e f u n c t i o n ,
fy r^
= .5
z /X jj
=
.3 1 7
( 2.
. v e r s u s n o rm a liz e d , tim e ,
( c o ^ z / c ^ = 1)
ccurve
u rv e
f o r th e
P
curve A.
ra d ia n s
noxmalized tim e. Co t .
*
F3£ure_12.
f
■■
:
o
16 )
^ m ode.
-6 9 v e c to r.
(2 ,1 6 )
T h is i s p a r t i c u l a r l y i n t e r e s t i n g f o r th e c a se ( f ^ / f ^ ) < 1
s i n c e , a s th e e v a n e s c e n t wave i s s e t up i n th e c u t - o f f g u id e so
t h e r e m ust he an e n e rg y flo w i n t o th e g u id e h u t th e pow er flo w
g iv e n hy th e tim e a v e ra g e P o y n tin g v e c t o r i s z e ro f o r th e s te a d y
s ta te fie ld s *
The in s ta n ta n e o u s P o y n tin g v e c t o r , w hich w i l l he
r e f e r r e d to a s th e P o y n tin g v e c t o r , P , i s g iv e n hy th e e x p re s s io n
P = Re (C
6 ^ ^ and
1
)
. Re ^ ^ t r ) '
th e p e r p e n d ic u la r v e c to r s
The tim e a v e ra g e P o y n tin g v e c t o r i s g iv e n hy
Re
A g a in , th e q u a n t i t i e s f ^ ( z , t ) and f^CiiS^t) a re ta k e n to
r e p r e s e n t th e t r a n s v e r s e m ag n e tic and e l e c t r i o f i e l d s so t h a t th e
P o y n tin g v e c t o r i s g iv e n hy th e p ro d u c t o f th o s e two q u a n t i t i e s .
However, i t w i l l he em phasized t h a t f o r th e com plete e x p r e s s io n ,
th e m u ltip ly in g c o n s ta n ts o f e q u a tio n s ( 2 , 1 4 * 1 ,2 ,3 ) must he u s e d ,
P ig u re 1 3 ( 7 0 ) shows a g ra p h o f th e P o y n tin g v e c t o r v e r s u s
n o rm a liz e d tim e , w ^ t , f o r ( f ^ / f ^ ) = ,3 and z / X ^ = o , . 0317; and
. 317*
Im m e d ia te ly a f t e r th e a r r i v a l o f th e w a v e fro n t, th e i n i t i a l
en erg y f l u x i s p o s i t i v e and t h i s p o s i t i v e e x c u r s io n h a s a h i g h e r
peak v a lu e th a n f o r l a t e r tim e s ,
i n i i l s t th e c u rv e s c o rre s p o n d in g to z /X ^ = o and ,0 3 1 7
n o t show th e r e g u l a r b e h a v io u r o f th e s te a d y s t a t e P o y n tin g v e c t o r ,
exp (-2 ct z) oos ( u i ^ t ) s i n ( W ^ t), t h e i r i r r e g u l a r i t i e s a re
n o t so g r e a t a s th o s e shown i n th e curve f o r ( z / X ^ ) = ,317*
z
Per
/ = o and , 0317; t h e 'p a t t e r n o f two p o s i t i v e and tv/o n e g a tiv e
e x c u rs io n s d u rin g a c y c le h a s h een e s t a b l i s h e d a lth o u g h t h e i r
p e ak s a re n o t e q u a l and occupy u n e q u a l le n g th s o f tim e b e in g
s l i g h t l y g r e a t e r th a n o r l e s s t h a n a q u a r t e r c y c le (A{w^t) = 1 ,3 7 )*
- 70F ig u re 1 3 .
( 2. 16)
T h e G ra p h o f t h e i n s t a n t a n e o i i s P o y n t i n g v e c t o r . P .
( z ,t) )
v e r s u s n o rm a liz e d tim e ,
f o r th e
V ^ c = "5"
C u rv e
■
' ' ■"
n
.5
A.
n
.3
.2
.1
-.1
—. 2
- .3
Figure 13
n o rm a liz e d tim e ,
(u ^ t.
(z/ X )
0
0
.0 3 1 7
.3 1 7
(w z / c )
0
0
m o d e,
- 71-
(2 .1 6 )
I n th e p re v io u s s e c t i o n , i t was n o te d t h a t th e m ag n e tic f i e l d
h ad re a c h e d th e s te a d y s t a t e f o r w ^t > 30 ( f i g u r e 7 ( 5 7 ^ w h i l e
th e e l e c t r i c f i e l d ( f i g u r e 8 (5 8 )) had n o t .
I t s irre g u la ritie s
g iv e r i s e t o th e i r r e g u l a r i t i e s i n th e P o y n tin g v e c t o r .
T h is
r e f l e c t i o n i n t o th e P o y n tin g v e c t o r o f th e d e v ia ti o n s from
s te a d y s t a t e o f th e e l e c t r i c f i e l d i s even more o b v io u s f o r
( z /X ^ ') = . 317, where th e r e i s no ap p ea ra n c e o f th e r e g u l a r
b e h a v io u r o f t h e s te a d y s t a t e f o r W ^t < 25.
N e x t, th e e f f e c t o f v a x y in g th e r a t i o o f th e c a r r i e r and
c u t - o f f f r e q u e n c ie s f o r g iv e n ( W z /c ^ ) (= 2 f r ( f ^ / f ^ ) z /
w i l l be i l l u s t r a t e d .
n o rm a liz e d tim e f o r
and
f / f ^ = .5 5 (
The g rap h o f th e P o y n tin g v e c t o r v e rs u s
tO z /o = . 1 , f / f
= ,7 5 (z /X
= .0211)
0
0
o c
o
= . 0167) a re g iv e n i n f^igure 14(72 ) .
U n lik e th e b e h a v io u r o f t h i s v e c t o r f o r f ^ / f ^ = .5^ w ^ z/c^ = .1
(z/X ^ = . 0317) j w here th e i n i t i a l p e a k energy f l u x im m e d ia te ly
f o llo w in g th e a r r i v a l o f th e w a v e fro n t was g r e a t e r th a n f o r
l a t e r tim e s , t h i s g r e a t e s t p o s i t i v e p eak i s fo u n d i n th e seoond
p o s i t i v e e x c u rs io n f o r ( f ^ f ) =; .7 3 and i n th e n i n t h f o r
( V ^ o ) = . 95.
I t i s n o t in te n d e d to make a com parison o f th e r e s u l t s o f
t h i s t h e o r e t i c a l i n v e s t i g a t i o n w ith th o s e o f c h a p t e r s th r e e an d
f o u r , w hich r e l a t e to an e x p e rim e n ta l i n v e s t i g a t i o n .
However,
v a lu e s o f th e p a ra m e te r ( f ^ / f ^ ) w i l l be ta k e n from e x p e rim e n t.
P o r exam ple, i n a p a r t i c u l a r e x p e rim e n t, a s i g n a l o f c a r r i e r
fre q u e n c y éUHz was p a s s e d th ro u g h a s h o r t le n g th o f g u id e o f
c u t - o f f fre q u e n c y 6*56(tHz .
Thus f ^ / f ^ = .9 0 3 and u )^ z/c = 1
( 2 . 16 )
-72F ig u re 14.
c u rv e
^
A
T he G r a ^ o f t h e P o y n t i n g v e c t o r ,
(w z / c ) = .1
o
o'
f /f
= . 75
f y f ° = .95
f ^ ( z .t) .
°
* v e rs u s n o m a liz e d tim e .
c*\ t ,
0
f o r th e
node,
0
z / \ = .0211
z /V ° = . 0 1 6 7
S
.2
.1
g
a m p l it u d e
a m p litu d e
- .1
F ig u r e
14.
n o rm a liz e d tim e ,
'
co t .
o
-7 3 -
( 2. 16)
when z = «Seras from th e b e g in n in g o f th e c u t - o f f s e c t i o n ,
Wo^/C q ~
and ,2j, z =: «08 and *l6cms r e s p e c t i v e l y .
For
The g ra p h
o f th e P o y n tin g v e c t o r v e rs u s n o rm a liz e d tim e f o r th e s e p a ra m e te rs
i s p r e s e n te d i n f i g u r e 15(74 )o
F i r s t l y , i t w i l l b e n o te d t h a t f o r U i^z/c^ = «1 (a n d in d e e d ,
.2 and 1) th e h ig h e s t p e ak v a lu e o c c u rs i n th e f i f t h p o s i t i v e
e x c u sio n after* th e w a v e fro n t a r r i v a l .
I t h a d b een n o te d p r e v io u s ly
t h a t t h i s o c c u r re d i n th e seco n d and n i n th p o s i t i v e e x c u r s io n s ..fQ*r
f ^ / f ^ = ,7 5 e.nd .9 5 r e s p e c t i v e l y f o r w ^ z /c ^ = ,1 (s e e f i g u r e 14) .
I n th e s e t h r e e exam ples, .75^ .9 0 5 and ,9 5
f o r cu^z/c^
= .1 ^ th e
n e t en erg y f l u x from th e tim e o f a r r i v a l o f th e w a v e fro n t to t h i s
h ig h e s t e n erg y f l u x p eak i s c e r t a i n l y p o s i t i v e and t h e r e f o r e
fo rw a rd i n t o th e g u id e .
S e c o n d ly , th e c u rv e s i n f i g u r e 15 i n d i c a t e th e o v e r a l l
re d u c tio n i n en erg y f l u x a s th e d i s t a n c e , z , i s in c r e a s e d an d
th e s te a d y s t a t e a m p litu d e s o f t h e P o y n tin g v e c t o r , p r o p o r t io n a l
to e x p ( - 2 a z ) , a re d e c re a s e d .
C om parison w ith f i g u r e 1 3 (7 0 ) f o r f f ^
= .5 ,
s ,1 ( z / X ^ - , 0317) shows t h a t t h e te n d e n c y to s te a d y s t a t e
a p p e a rin g t h e r e i s n o t a p p a r e n t f o r i o ^ t < 2 5 i n f i g u r e 15 f o r
( f o / f c ) = . 9 0 5 , W ^z/c = , 1 ,
I t Vfa 8 f o r t h i s r e a s o n t h a t th e
v a lu e f ^ / f ^ = .5 was chosen so t h a t th e s te a d y s t a t e o r a te n d e n c y
to i t w ould become a p p a re n t o v e r th e range o f w ^ t which c o u ld
be r e a s o n a b ly p l o t t e d .
2 .1 7
V a r i a t i o n _pf th e t r a n s v e r s e f i e l d com ponents w ith d i s t a n c e ,
(z /X
) f o r g iv e n v a lu e s o f ( f / f ) and t im ej (<i- t ) ,
In t h i s th e o r e tic a l in v e s tig a tio n f o r f ^ / f ^ < 1 , th e re has
Figure 15.
The Graph of the Poyntir^ vector,
-74(2.17)
z,t^ .f^ (z .t). versus normalized t im e ^ ^t, for the_H_^_mo^.
.1
i
-.1
C urve
cOo Z/C q
n
.1
.0 1 7 5
.2
.0 3 5
1
.1 7 5
n
2
4
F ig u re 1 5 .
n o rm a liz e d tim e ,
-7 5 (2 .1 7 )
b e e n a s h i f t o f em phasis from th e v a r i a t i o n o f one com ponent w ith
tim e ( a s i n p r e v io u s i n v e s t i g a t i o n s f o r f ^ / f ^ < 1 o r f ^ / f ^ > 1) to
th e v a r i a t i o n o f a p a i r o f t r a n s v e r s e com ponents w ith tim e and
c o n s e q u e n tly to a s tu d y o f th e P o y n tin g v e c t o r .
I n t h i s s e c tio n ,
t h e r e w i l l be a f u r t h e r s h i f t o f em phasis to a c o n s i d e r a ti o n o f
th e v a r i a t i o n o f th e t r a n s v e r s e f i e l d components w ith d i s t a n c e ,
z , a lo n g th e g u id e ,
f o r th e s te a d y s t a t e f i e l d s th e a m p litu d e s
decay e x p o n e n tia lly w ith th e d i s t a n c e , z , th u s
^ ^ r “ ^o
-CCz) s in ( W ^t)
and
e x p (-a z ) c o s ( w ^ t) ,
I n f i g u r e 1 6 (7 6 ) ; th e m ag n etic f i e l d h a s b e e n p l o t t e d a g a i n s t
th e n o rm a liz e d d i s t a n c e , ( z /X ^ ) , f o r th e n o rm a liz e d tim e s ,
oj^t = 3 .9 2 7 and 8 .6 4 ( t h a t i s , 5 /8 and l f c y c le s o f th e c a r r i e r
w av e).
The v /av efro n t h a s re a c h e d th e p o s i t i o n s , z / X ^ = 1 * 23
and 2 .7 5 r e s p e c t i v e l y ( f ^ / f ^ = . 3 ) .
The b ro k e n c u rv e s i n d i c a t e
th e s te a d y s t a t e m ag n e tic f i e l d f o r W^t = 3*927 and 8 . 6 4 .
It
can be seen t h a t f o r d i s t a n c e s z / X ^ < ,4^ th e cu rv e f o r
w ^ t = 3*93 c o in c id e s w ith th e s te a d y s t a t e c u rv e .
For w ^t =
8 . 6 4 , t h i s r e g io n o f s te a d y s t a t e m ag n e tic f i e l d e x te n d s f u r t h e r
i n t o th e g u id e , a s e x p e c te d up to a p p ro x im a te ly ( z / X^) -• *7»
f i g u r e 1 7 (7 7 ) c o n ta in s a g rap h o f th e t r a n s v e r s e e l e c t r i c
f i e l d v e rs u s th e n o rm a liz e d d i s t a n c e , ( z / X ^ ) , f o r w ^ t = 3*93
and 8 .6 4 and (f^ /f^ )r= ,3„
H e re , t h e r e i s a l a r g e d i f f e r e n c e
b etw een th e s e c u rv e s and th e s te a d y s t a t e curve (b ro k e n l i n e ) .
I t m ust be rem embered t h a t i t i s th e form o f th e t r a n s v e r s e
m ag n e tic f i e l d w hich i s s p e c i f i e d a t z = o ( th e u n i t s te p
m o d u lated c a r r i e r w a v e).
The o t h e r f i e l d com ponents, in c lu d in g
th e t r a n s v e r s e e l e c t r i c f i e l d , a re d e r iv e d from i t .
W ith th e
F ig u r e l 6 .
-7 6 The G raph o f th e t r a n s v e r s e m a g n e tic f i e l d co m p o n en t,
(z /\
, f o r th e
m ode.
V ^ o = -5-
( 2 .1 7 )
v e r s u s n o r m a liz e d d i s t a n c e .
c*>^t = 3 * 9 3 a n d 8 ,6 lf
.8
.6
C u rv e
u t.
— 0—
fu n c tio n
r
3 .9 3
f o ( '. t )
1’ .
3 .9 3
s te a d y s t a t e
/V
8 .6 4
8 .6 4
ste a d y s t a te
" .8
-.6
.4
.4
.2
a m p litu d e
a m p litu d e
0
-.2
- .4
- .6
n o rm a liz e d d i s t a n c e , ( z / X ^ ) .
F i g u r e 16
( 2 .1 7 )
-7 7 T he G ra p h o f t h e t r a n s v e r s e e l e c t r i c f i e l d . f ^ ( z , t ) , v e r s u s iio rm alizecL d i s t a n c e . z / \
F ig u re 17 .
= .5 .
w ^t=
, f o r th e
m ode.
3 . 9 3 a n d 8 .6 4
C u rv e
.2
fu n c tio n
Co
3 .9 3
3.93
f 2 (z,t)
ste a d y s t a te
.6 4
8 .6 4
s te a d y s t a te
^ 2( 2,^ )
.1
a m p litu d e
a m p litu d e
0
-.1
—.2
-.3
-.4
i
FiRure 17.
n o rm a liz e d d is t a n c e ,
(z /V ^ ).
-7 8 -
(2 .1 7 )
t r a n s v e r s e e l e c t r i c f l e l d ^ a s f i g u r e 1 0 (6 3 ) shows f o r
z /X ^ = o t h e r e i s no a p p ea ra n c e o f th e s te a d y s t a t e f o r
W ^ t< 5 0 ; a lth o u g h t h e r e i s some te n d e n c y to i t s r e g u l a r
b e h a v io u r.
So f o r th e s e r e l a t i v e l y sm a ll v a lu e s o f tim e ,
W^t = 3*93 and 8 .6 4 f o r z /X
o , th e l a r g e d i f f e r e n c e betw een
th e e l e c t r i c f i e l d and th e ste a d y s t a t e i s n o t u n e x p e c te d .
T h is l a r g e d e v i a t i o n from s te a d y s t a t e i s r e f l e c t e d i n
th e P o y n tin g v e c t o r , w hich i s i l l u s t r a t e d i n f i g u r e 1 8 ( 7 9 )
a s a f u n c t i o n o f th e n o rm a liz e d d i s t a n c e , z / X f o r
^ '
o
w t =
o
3 .9 3 and 8 .6 4 ( f u l l c u rv e s ) w ith th e s te a d y s t a t e c u rv e s a ls o
(b ro k e n c u r v e s ) .
P o r v a lu e s o f z / X ^ s l i g h t l y s m a lle r th a n th e
v /a v e fro n t p o s i t i o n , th e P o y n tin g v e c t o r i s alw ays p o s i t i v e .
These p o i n ts c o rre s p o n d t o p o s i t i o n s w hich th e w a v e fro n t h a s
j u s t p a s s e d o v e r.
I t can be se e n i n f i g u r e s 1 3 (7 0 )p 14( 7 2 ) ,
1 5 (74 ) t h a t j u s t a f t e r th e a r r i v a l o f th e w a v e fro n t th e P o y n tin g
v e c t o r was p o s i t i v e , i n d i c a t i n g an e n e rg y f l u x fo rv /a rd i n t o th e
g u id e , w hich i s e x p e c te d on p h y s ic a l g ro u n d s,
2 ,1 8
D is c u s s io n and C o n c lu sio n s
The c o m p u ta tio n s and g ra p h s o f a p a i r o f t r a n s v e r s e f i e l d
com ponents and th e P o y n tin g v e c t o r a s f u n c tio n s o f tim e (w ^t)
and d is ta n c e (z /X ^) f o r a g iv e n r a t i o ( f ^ / f ^ ) co m p lete th e
t h e o r e t i c a l i n v e s t i g a t i o n o f th e p ro p a g a tio n o f a u n i t s t e p
m o d u lated c a r r i e r wave i n a w aveguide o r p lasm a .
c o m p u ta tio n s, th e
p a rtic u la r.
P o r th e s e
mode f o r w aveguide was c o n s id e re d i n
However, th e t h e o r e t i c a l r e s u l t s o f s e c t i o n (2 ,1 3 )
may be a p p li e d to T .E , o r T.H.
modes w here f ^ ( z = o , t )
( th e
one component s p e c i f i e d a t th e o r i g i n ) v d .ll r e p r e s e n t th e
F i/^ u r e
( 2 . 18)
18 .
T he G rap h o f
f^ (z,t).
fgfzpt)
d is ta n c e ,
— — —— '
w
t
tn o P o y n tin g v e c t o i; P ,
z /X
.
O
versu s
f
/ f
O
0
n o r m a liz e d
=
,5
= 3 « 9 3 ancL 8 . 6 4 .
TTT'nïïI
i ;
i!i
I
i
I! !
#
III
C urve
fu n c tio n
3.93
3 .9 3
P
ste a d y
8 .6 4
Hi
8 .6 4
P
ste a d y
ia iiiii
F ig u r e
n o r m a liz e d d i s t a n c e ,
sta te
z /X
sta te
'*'80""'
(2a 18)
t r a n s v e r s e m ag n e tic o r e l e c t r i c f i e l d r e s p e c t i v e l y .
C o n v e rse ly ,
f g f z f t ) , th e component d e riv e d from f ^ ( o , t ) w i l l r e p r e s e n t th e
t r a n s v e r s e e l e c t r i c o r m a g n e tic f i e l d .
However, i n t h i s f u r t h e r
d i s c u s s i o n th e f u n c t i o n s f^ (% ,t) and f 2( z , t ) vd-ll r e t a i n th e
c o n n o ta tio n o f th e p r e v io u s s e c t i o n s .
The o u ts ta n d in g p o i n t to emerge from th e s e c o m p u ta tio n s ,
i s t h a t th e d e riv e d t r a n s v e r s e e l e c t r i c f i e l d component ta k e s a
v e iy g r e a t tim e to r e a c h s te a d y s t a t e ( f o r exam ple, see f i g u r e
10(63) f o r z / X ^ = o ) ,
The o r i g i n o f t h i s b e h a v io u r may be
t r a c e d to th e i n i t i a l c o n d itio n s i n th e tim e dom ain.
I n th e s e t t i n g up o f th e p ro b lem , i t was assum ed t h a t th e
o n ly i n i t i a l c o n d itio n i n th e tim e domain o r e x c i t a t i o n a t
z = o was th e t r a n s v e r s e m ag n etic f i e l d (T ,E . wave
H ^ ^ ),
B ^ ( o ,s ) , and th e o t h e r , nam ely B ^ ( o ,s ) , was z e r o .
T h e re fo re
from page A ,5 o f A ppendix A, i t can be se e n t h a t
( o ,s ) =
Eg ( o ,s ) = o so t h a t t h e r e i s no so u rc e o f e x c i t a t i o n o f th e
tra n s v e rs e e le c t r i c f i e l d .
T h is i s a se v e re r e s t r i c t i o n on th e
e x c i t a t i o n o r s e t t i n g up o f th e s i g n a l i n th e g u id e a t z = o ,
A more p r a c t i c a l a p p ro a c h to th e s e t t i n g up o f a wave a t z = o
I
f o r a T .E , wave f o r i n s t a n c e , w ould in v o lv e b o th
and
b e in g non z e r o .
A f u r t h e r p o in t w i l l be m en tio n ed a b o u t th e c o n d itio n
t
B;, (o,s)::jr o ,
B -,(o ,s ) = o ( o r E .( o ,s ) = E ^ ( o ,s ) = o ) .
y
A g la n c e
a t page A.A o f A ppendix A w i l l i n d i c a t e t h a t i f B^ ( o , s )
o,
th e n E ^ ( o ,s ) and E2( o , s ) v iiio h a re p r o p o r t io n a l to B^ ( o ,s ) a re
a ls o n o t z e ro a t z = o .
However, th e e x p re s s io n s c o n ta in e d i n
p a g e s A.1 to A,A o f A ppendix A a re f o r th e waves i n th e p o s i t i v e
*
z d ire c tio n
o n ly .
-
*
8
1
(
2
©
18)
I f th e waves i n th e n e g a tiv e z d i r e c t i o n a re
in c lu d e d a ls o (a n d i n t h i s p roblem t h e r e i s no s t i p u l a t i o n t h a t
th e s e waves sh o u ld n o t be in c lu d e d - i t i s m e re ly c o n v e n ie n t to
c o n s id e r th e d i r e c t waves o n ly ) th e n a t z = o , E ^ (o ,s ) and E^Co^s)
a re p r o p o r t io n a l to
( s / ( s ^ + 1 ) ^ (e x p (~ z (s ^ + ^ ) ^ / ' c)
- 6 x p (z (s ^ + 1 )
c) )
Z =3 o
which i s zero»
A n o th e r im p o r ta n t and i n t e r e s t i n g p o in t h a s em erged from
c o n s i d e r a ti o n o f th e P o y n tin g v e c t o r s a f u n c t i o n o f n o rm a liz e d
k
tim e , Cjü^t, f o r a g iv e n d i s ta n c e a lo n g th e g u id e .
Im m e d ia te ly
fo llo w in g th e a r r i v a l o f th e w a v e fro n t th e P o y n tin g v e c t o r malces
a p o s i t i v e e x c u r s io n above th e a x i s .
♦O317 (c u ^ z /c
= .1 ) , f^ /f ^ =
Por z /
= o and
(fig u re l3 (7 0 )) th is f i r s t
p o s i t i v e e x c u r s io n h a s a p e a k v a lu e g r e a t e r th a n a l l su b se q u e n t
p o s i t i v e o r n e g a tiv e e x c u r s io n s .
F o r o t h e r v a lu e s o f th e r a t i o
( f ^ / f ^ ) , t h i s f i r s t e x c u r s io n does n o t have th e g r e a t e s t p e ak
v a lu e b u t i s fo llo w e d by p o s i t i v e and n e g a tiv e e x c u r s io n s , b o th
in c r e a s in g i n p e ak v a lu e .
The l a r g e s t p o s i t i v e p eak o c c u rs
l a t e r f o r i n c r e a s i n g v a lu e s o f th e r a t i o
w ^ z /c ^ = c o n s t a n t .
F o r exam ple, i t o c c u rs i n th e seco n d f o r f ^ / f ^ = .7 5 and i n th e
n i n t h f o r f ^ / f ^ = *95^w^z/o
= .1 f o r b o th ^ in f i g u r e 1 4 (7 2 ) and
th e f i f t h f o r f ^ / f ^ = .9 0 5 i n f i g u r e 1 5 (7 4 ) , w ith w ^z/c^ = . 1 .
U n t i l th e o c c u rre n c e o f t h i s maximum peak v a lu e , th e n e t e n e rg y
f l u x i s p o s i t i v e ( i n d i c a t e d q u i t e c l e a r l y by th e g r e a t e r a r e a
u n d e r th e cu rv e above th e a x is th a n below i t ) ,
F o r tim e g r e a t e r
th a n t h i s , i t h a s n o t b e e n p o s s i b l e to f i n d such a s p e c i f i c
tre n d .
However, th e r e i s a te n d e n c y f o r th e p o s i t i v e and n e g a tiv e
- 82-
( 2. 18)
e x c u rs io n s to e q u a liz e i n a m p litu d e and th e tim e each o c c u p ie s , as
th e s te a d y s t a t e i s s e t up,
F o r th e s te a d y s t a t e , th e r e a re two
p o s i t i v e and two n e g a tiv e e x c u rs io n s i n each c y c le , e q u a l i n
a m p litu d e , th e r e b y g iv in g a z e ro tim e a v e ra g e P o y n tin g v e c t o r .
T h u s, t h i s t h e o r e t i c a l i n v e s t i g a t i o n o f th e tim e d ep en d en t
problem o f th e e v a n e sc e n t wave i n a w aveguide o r plasm a i s
c o n clu d ed .
I t i s n o t a com prehensive s o l u t i o n , i n th e sense
t h a t th e i n i t i a l tim e c o n d itio n s o r so u rc e o f e x c i t a t i o n a t
z = o a re v e ry r e s t r i c t e d .
N e v e r th e le s s , a t t e n t i o n h a s b e e n
drawn to th o s e a s p e c ts o f th e p ro b lem , nam ely a p a i r o f t r a n s v e r s e
com ponents and th e i n s ta n ta n e o u s P o y n tin g v e c t o r , w hich a re
p a r t i c u l a r l y r e l e v a n t to th e e v a n e sc e n t wave.
(3.0)
CHAPTER 5 .
A TIME - DEPENDENT STUDY QE UTOSTR/lTED TOTAL REFLECTION*.
3 .0
In tro d u c tio n
I n t h i s c h a p te r , i t i s p ro p o se d to c o n s id e r t h e o r e t i c a l l y
th e a p p l i c a t i o n o f a s i g n a l to a w aveguide system c o n s i s t i n g o f
a s e c t i o n o f *c u t-o ff* g u id e , sandw iched betw een tv/o i d e n t i c a l
p r o p a g a tin g w av eg u id es, a s shown i n f i g u r e 2(1?X *c u t - o f f ’ o r
p r o p a g a tin g w ith r e s p e c t to th e c a r r i e r fre q u e n c y o f th e s i g n a l ) .
F o r th e sake o f c o n v e n ie n c e , ci G a u ssia n m o d u la tio n e n v elo p e w i l l
be c o n s id e re d i n th e f i r s t i n s t a n c e .
The a n alo g y b etw een t h i s
and th e quantum m e c h a n ic a l tu n n e l e f f e c t , a lr e a d y d e s c r ib e d i n
C h a p te r 1 f o r s te a d y s t a t e c o n d itio n s , w i l l be drawn i n some
d e t a i l a s i t i s f e l t t h a t t h i s w i l l g iv e some h e lp i n c o n s id e r in g
th e w aveguide system .
The t h e o r e t i c a l l y p r e d i c t e d r e s u l t s w i l l be re v ie w e d f o r th e
p o s s ib i l i ty o f p r a c tic a l a p p lic a tio n .
I t w i l l be p o s t u l a t e d t h a t ,
u n d e r v e ry l i m i t e d c o n d it i o n s , th e y can be u s e d i n t h i s c o n n e c tio n ,
3.1
The *tu n n e l e f f e c t* - s te a d y s t a t e c o n s id e r a tio n
H u p e r t( ^ ^ ) '( ^ ) and Campi and H a r r i s o n ^ h a v e p r o v id e d a
c i r c u i t th e o r y i n t e r p r e t a t i o n o f th e tu n n e l e f f e c t , u n d e r s te a d y
s t a t e c o n d itio n s and t h e i r r e s u l t s w i l l be b r i e f l y en u m e rated .
The b e h a v io u r o f t h e p a r t i c l e i s g o v e rn e d by th e Shrb*dinger
e q u a tio n
jh . d
/
d t =
-(-h ^/'2 m )
+
V ( z )g j
f o r th e m o tio n o f th e p a r t i c l e i n th e d i r e c t i o n o f p o s i t i v e z .
The com plex f u n c tio n
i |; ( z ,t ) i s d e fin e d su ch t h a t
re p re s e n ts
-8 4 th e p r o b a b i l i t y d e n s ity o f th e p a r t i c l e b e in g a t z , t .
o f th e v a r i a b l e s y i e l d s ip ( z , t ) =
C exp ( - j E t / h ) and
where
^ g (t)
( 3 .1 )
S e p a ra tio n
w here
=
^ ^ ( z ) i s th e s o l u t i o n o f th e e q u a tio n
= 2m (E-V ) / *h^
^o o /
and E ^ a n d 'h a re th e e n e rg y o f th e p a r t i c l e and P lan o k * s c o n s ta n t
re s p e c tiv e ly .
I t i s h e re t h a t th e s i m i l a r i t y to th e w a v e e q u a tio n
can be n o te d .
The wave f u n c tio n s f o r th e t h r e e r e g io n s a re a s
f o llo w s : ~
R eg io n I
R egion I I
R egion I I I
=
A .e x p (jk ^ z ) + B .e x p ( -jk ^ z )
= D .exp(-kgZ ) + B .e x p (k g z )
=0.
( s e e f i g u r e 3(i8))*
.(jk ^ z )
The im p o r ta n t p o in t to n o te i s th e change o f
w a v e fu n c tio n a s th e p a r t i c l e e n t e r s th e b a r r i e r r e g io n , from an
im a g in a iy p ro p a g a tio n c o n s t a n t , y
a s i n th e c a se o f th e c u t - o f f w av eg u id e.
to a r e a l one
Y = -k g ,
The t a b l e on pâg;e:
a f f o r d s e a sy com parison o f t h e v a r io u s a n a lo g o u s q u a n t i t i e s .
(85)
B oth
th e ’’tu n n e lin g e f f e c t s ” r e q u i r e th e t o t a l w a v e fu n c tio n i n th e b a r r i e r
r e g io n , ip =
.e x p ( - k 2z)+ E ,e x p (k g z ) to o b ta in a n o n -z e ro
p r o b a b i l i t y d e n s ity c u r r e n t o r tim e a v e ra g e P o y n tin g v e c t o r ,
3 ,2
The *tu n n e l e f f e c t* - th e tim e d ep en d en t problem
To a c c o u n t f o r th e tim e sp e n t i n tu n n e lin g th ro u g h th e b a r r i e r ,
th e t r a n s m is s io n o f a vrave p a c k e t o r s i g n a l i n th e e le c tro m a g n e tic
case m ust be c o n s id e re d .
When th e i n c i d e n t p a c k e t c o l l i d e s w ith th e
b a r r i e r , i t s p l i t s i n t o a t r a n s m i t t e d and r e f l e c t e d p a c k e t,
M aoooll
( 35')
s t a t e d t h a t th e p e ak o f th e t r a n s m i t t e d p a c k e t le a v e s
•8 5 -
T unnel E f f e c t
C h a ra c te ris tic
Im pedance
1 = j/Y
Y = (ôA).
P ro p a g a tio n
C o n sta n t
(3.2)
T .E ,^U, ^1 W aveguide
T| = j w n / Y
Y = (j/o ).
.(u)^ -
O p e ra tin g fre q u e n c y
w
C u t- o f f fre q u e n c y
f o r w > o/o fre q u e n c y
w
< 0/0 fre q u e n c y
vyb
Y im a g in a ry ,
Y r e a l , n e g a tiv e ,
E le c tric f i e l d
M ag n etic f i e l d
0
$ (a )
jy; (z )
S q u are o f modulus
4'
( p r o b a M |i t y
P r o b a b i l i t y d e n s ity
c u rre n t /
P o y n tin g V e c to r
KdCqT t | / h / 2m)
u)
kgO
T] r e a l
T] im a g in a ry
C(s)
H ( z ) tr a n s v e r s e
£^(2)
Re(&£
)
T ab le o f a n a lq ^ o u s q u a n t i t i e s f o r th e quantim neohariL cal
and w aveguide t u n n e l e f f e c t s ( a f t e r H u p e rt^ ^ ^ )),
- 86-
( 3 . 2)
th e b a r r i e r " a t a b o u t th e i n s t a n t " th e p eak o f th e i n c i d e n t
p a c k e t a r r i v e s a t th e b a r r i e r so t h a t t h e r e i s no a p p r e c ia b le
d e la y ,
H a r t m a n ^ a n d P o rla h ii and M in n a ja ^ ^ ^ ^ , i n c o n s id e r in g th e
p a ssa g e o f a one d im e n sio n a l G-aussian wave p a c k e t th ro u g h a
r e c t a n g u l a r p o t e n t i a l b a r r i e r , have exam ined th e problem more
d e e p ly .
U nder c e r t a i n c o n d it i o n s , th e y have fo u n d tr a n s m is s io n
i n t e r a c t i o n tim e s .
T h e ir m ethod o f s o l u t i o n c o n s i s t s i n f i n d i n g th e s te a d y
s t a t e s o l u t i o n s , m u ltip ly in g t h i s by th e G-aussian w e ig h tin g
2 /
2
f a c t o r , G(^k) = e x p ( - ( k - k j / 2 Ak ) and th e n i n t e g r a t i n g o v e r a l l
th e p o s s ib l e s t a t e s .
W ith th e p re v io u s n o t a t i o n , a t y p i c a l
i n t e g r a l w i l l be
p+CX)
(p = C \
exp ( - ( k - k
/ 2 A k^ - j w t + ‘jkz) R (k ,a )d k
°
. .....( 3 .2 .1 )
T(lc,a)
-00
where C i s a c o n s ta n t,R ( k ,a ) and T ( k ,a ) a re r e f l e c t i o n and
tr a n s m is s io n c o e f f i c i e n t s and a re f u n c tio n s o f th e vfavenumber, k ,
and b a r r i e r t h ic k n e s s , a .
The f u n c tio n G(k) i s th e F o u r i e r
tr a n s fo rm i n t o k sp ace o f t h e i n i t i a l c o n d itio n i n th e tim e
domain w h ile
expQkz) ,R (k ,a ) a re th e s te a d y s t a t e t r a n s f e r
T (k ,a )
o f th e b a r r i e r r e g io n .
f u n c tio n s
The i n t e g r a l o f ( 3 .2 .1 ) i s th u s th e
F o u r ie r i n v e r s i o n i n t e g r a l and so i s com parable w ith th e i n v e r s i o n
i n t e g r a l o f ( 2 ,7 ,1 ) s in c e th e f u n c t i o n F (p ) th e r e i s a ls o th e
tra n s fo rm (L a p la c e ) o f th e i n i t i a l c o n d itio n i n th e tim e dom ain.
-8 7 -
( 3 .2 )
However^ th e t r a n s f e r f u n c t i o n o f th e system i s no lo n g e r
e]qp( Y s) b u t e x p (k z )
^ .
I t does n o t seem p o s s i b l e to e v a lu a te
i n t e g r a l s o f th e ty p e c o n ta in e d i n e q u a tio n ( 3 , 2 . 1 ) .
However,
i t i s hoped t h a t by i n v e s t i g a t i n g th e phase o f th e in te g r a n d b y
th e
s t a t i o n a r y p h a se m ethod and a ls o th e modulus o f
i t w i l l be p o s s i b l e to o b t a i n some i n s i g h t
th e i n te g r a n d ,
i n t o th e p h y s ic a l
b e h a v io u r o f th e sy ste m ,
3 .3
The S t a ti o n a r y Phase M ethod
The t y p i c a l i n t e g r a l i n ( 3 .2 ,1 ) i s o f th e form
Gr(j),exp
(-jK (y ))
dy
J'1
w here K(y) i s r e a l and c o n ta in s a l l th e r a p i d l y v a ry in g phase
te rm s w h ile &(y) i s a slo w ly v a ry in g f u n c tio n w hich may be com plex.
The p r i n c i p l e o f s t a t i o n a r y p h ase s t a t e s t h a t th e m ain c o n tr i b u ti o n
to th e i n t e g r a l comes from th e p o i n t s w here th e p h a se K(y) i s
s t a t i o n a r y , t h a t i s , w here
ô K (y ) y / d y = 0.
I n th o s e p la c e s
where K(y) v a r i e s r a p i d l y , e x p (~ jK (y )) o s c i l l a t e s much and th e
i n te g r a l i s p r a c t ic a l l y zero .
I n th e c ase o f th e w avepacket d e s c r ib e d by ( 3 .2 .1 ) f o r each
v a lu e o f tim e , t h e r e w i l l be one p o i n t where waves o f d i f f e r e n t k
do n o t te n d to i n t e r f e r e d e s t r u c t i v e l y .
At t h i s p o i n t th e r e w i l l
be a ran g e o f k where a l l waves have n e a r ly th e same phase and w i l l
in te r f e r e c o n s tru c tiv e ly .
_d__ ("ü)t + kz +
T h is p o in t i s found by p u t t i n g
m ( k ,a ) ) = o
8%:
t h a t i s , - 8w , t
3k
+ z +
6vp
3k
( k .a ) = o
’
-8 8 -
( 3 .3 )
w here tpg ^ ( k ,a ) i s th e p h ase o f t h e r e f l e o t i o n o r tr a n s m is s io n
00e f f i c i e n t , R o r T,
As a p o i n t o f i n t e r e s t , i f th e i n t e g r a l I can f u r t h e r he
e x p re s s e d as
& (y ), exp ( - j t ip (y )) dy
where t i s v e ry l a r g e and p o s i t i v e th e i n t e g r a l h a s th e
v a lu e (34)
1
G ( y J ( 2 t / t U " ( y j |, )® exp ( + j i t / 4 + t i p ( y ) )
I*
w here
(y )/d y
j ^ ^
= 0
T h is method i s c l o s e l y r e l a t e d to th e s a d d le - p o in t o r s t e e p e s t
d e s c e n t method u s e d i n C h a p te r 2 i n c o n n e c tio n w ith th e
e v a lu a tio n o f th e in v e r s e L a p la c e tr a n s fo rm s ,^
3 .4
The w aveguide " tu n n e l e f f e c t " *- s te a d y s t a t e c o n s id e r a tio n s
F o r th e v/aveguide sy stem o f f i g u r e 2 a ,
2h(17 ) th e system o f
e q u a tio n s to he s o lv e d w i l l now he s e t up.
I n view o f l a t e r p r a c t i c a l w ork, th e dom inant o r T ,E ,^Q mode
i n r e c t a n g u l a r w aveguide i s c o n s id e re d .
Where th e change from
im a g in a ry to r e a l p r o p a g a tio n c o n s ta n t o c c u rs i n p a s s in g from one
d i e l e c t r i c medium to a n o th e r , th e n no h i g h e r o r d e r modes w i l l he
s e t up.
On th e o t h e r h a n d , when t h i s change i s b ro u g h t ah o u t hy a
change i n th e b ro a d d im e n sio n o f th e w av eg u id e, th e n h i g h e r o r d e r
modes w i l l he s e t u p ^ ^ ^ \
However, f o r s i m p l i c i t y t h i s a n a l y s i s
c o n s id e r s th e dom inant mode o n ly s in c e i n th e p ro p a g a tin g r e g io n ,
th e dom inant mode c a r r i e s a l l th e pow er and i t w i l l he assum ed
-8 9 "
(3 ,4 )
t h a t i n th e c u t - o f f r e g i o n , m ost o f th e pow er i s c a r r i e d by th e
lo w e st o r d e r mode.
Campi and H a r r i s o n ^ h a v e a ls o c o n s id e re d
th e dom inant mode o n ly i n t h e i r e x p e rim e n ts on th e w aveguide
a n alo gue o f t u n n e lin g w ith th e sudden r e c t a n g u l a r b a r r i e r and
th e more slo w ly v a iy in g h y p e r b o lic b a r r i e r .
T h u s, f o r th e w aveguide system i n Y/hioh a s e c t i o n o f g u id e ,
w ith p r o p a g a tio n c o n s t a n t ,
i s sandvfiched betw een two i d e n t i c a l
g u id e s o f p r o p a g a tio n c o n s t a n t ,
, f o r a n g u la r fre q u e n c y ,
u ),
and th e TE^^ mode, th e system o f e q u a tio n s i s :
R egion I f o r z < o
E
=1
'
H
v
s in ( ir y /b ) j
/
(AoexpC-Y-i^) + Bo exp (y >,z ))«
s i n ('rry/b)
(A*exp(~Y^g) - Bo exp (Y^a)
R egion I I f o r o < .E <a,
jg)p,Tr
1
(ir(y -p ))
^2
H =j
y
~
j(D oexp(-Y p^) + E .e x p (Y p 2 ))
^ D o exp (-Y p^) ~ So exp (Y p^))
kj
b
R egion I I I f o r z > a
.(joopn* H
E
=
i — =--------\
H
(Cc.exp (« Y i^ ))
»
/ Y-i 'nr H
= ~ P ——
\
s i n (iry /b )
\
s i n ( i r y /b ) i (Caoxp ( - Y i^ ) )
^
/
From th e s e e q u a tio n s C/A and B/A may be found:
C/A
=
k-o exp (Yi&)»
. (exp (y 2® )(''+ Y i/Y 2 )(''+ Y 2^'V'|) + exp(-Y 2a ) ( l - Y i / Y 2 ) ( 1-Y2/ Y i ) )
«» O0 (ioAo ^)
(3 .4 )
-9 0 -
and
B/A
=
- Y g)(exp (y ^ a ) - exp ( - y ^ ) ) «
'(exp(Y2^)(Y -1 +
3*3
~
("^ 2 ^ )(Y 2 ~ Y ^)^) ^
«""« (3 « 4 .2 )
The S t a t i o n a r y P hase M ethod a p p lie d to th e w aveguide tu n n e l e f f e c t ,
I n th e f i r s t i n s t a n c e , th e i n i t i a l c o n d itio n i n th e tim e dom ain
V v ill
be a G-aussian a m p litu d e m o d u la ted c a r r i e r w ave.
The C o u r ie r
tr a n s fo r m te c h n iq u e i s u se d i n s t e a d o f th e L a p la c e one o f C h a p te r 2 ,
A G-aussian m o d u la tio n en v elo p e i s d i f f i c u l t to h a n d le w ith th e l a t t e r .
T h is i s b e c a u se th e G-aussian f u n c t i o n ( e x p ( ~ t ^ / a ^ ) e x i s t s f o r
~CD<t<oo vfhereas th e L a p la c e tr a n s f o r m range o f i n t e g r a t i o n e x te n d s
from t z z o t o t c o o .
A lso t h e r e i s
some m a th e m a tic a l s i m p l i f i c a t i o n
i n u s in g th e G -aussian f u n c t i o n a s i t s image f u n c t i o n i n th e fre q u e n c y
dom ain i s a ls o a G-aussian f u n c t i o n .
T hus, th e w avepacket o r s i g n a l
r e f l e c t e d from th e f r o n t fa c e o f th e b a r r i e r r e g io n and th e one
t r a n s m i t t e d th ro u g h th e b a r r i e r r e g io n a re g iv e n r e s p e c t i v e l y by
th e s e i n t e g r a l s ;
f\4-00
•CD
w ^ c ( Y. Yo
A
^ ^
a) e x p (j w t ) .d w
Ycv)
and
4-00
tJÜ- W )£ ( Y.
Yo
a) exp( j w t ) .d w
-C O
....................( 3 .5 .2 )
and C/A and B/A a r e g iv e n by e q u a tio n s ( 3 .4 .1 ) and ( 3 .4 .2 )
-9 1 2
Y '' =
2
and
-
2
y
W lie
,
= o r
( 3 .5 )
2
y / =
2
2
w |i&
-
, Yg
2
2
= k^^
2
-
l / j [i£ )
F o r f r e q u e n c i e s betw eenü^^ and
=
jP
and
Yg
?
in th is a n a ly s is .
and
Yg
B/A rs
w hich i s th e s i t u a t i o n o f m ain i n t e r e s t
F o r f r e q u e n c ie s g r e a t e r th a n
~ 0 'i^2*
^ Y^ = j
th e fo rm e r c ase
+ a ^ ) s in h ( a a ) ( ( /9 ^ - o :^ ) s in h (a a ) + 2joî/3 c o sh ( a a ) ) *
* ((j9^ - a ^ ) sin h '^ (a a ) +
and
oosh^ ( a a ) )
2
2
C/A = 2j(%j9 exp ( i/3 a ) ( ( a -/3 ) s in h (cca) « 2 ja/3 cosh ( a a ) ) ,
sin h ^ ( a a ) ^ h
0 ((
oosh^ (w a))
The ph ase o f th e in te g r a n d o f th e r e f l e o t i o n i n t e g r a l i s
= w t^
-
P
+
b (w , a)
= w t^
+
b ( u > ,a )
at z = o
and s i m i l a r l y o f th e tr a n s m is s io n i n t e g r a l
*^T ~
=
^ ^ T ** ^ ^T
(Jüt^ -
|3a
o(ciJja)
+
c(u>,a) a t z = a
where t^^ and t ^ a re th e r e f l e c t i o n and t r a n s m is s io n i n t e r a c t i o n
tim e s r e s p e c t i v e l y and
b ( u ,a ) = a r c t a n ( 2 a p c o s h ( a a) /
( (3^ - a ^ ) s in h ( a a) )
.( 3 .5 .3 )
and
c(ci>,a) = a r c t a n ( - ( P
2
2
- a )s in h (a a )
/
/
2ap
cosh ( a a ) ) -t# o » o s * (3 . 5 . ^)
P u ttin g th e f i r s t d i f f e r e n t i a l o f
and
w ith r e s p e c t to
(\)
e q u a l to z e ro to f i n d th e extremum o r s t a t i o n a i y p h ase p o i n t , th e
fo llo w in g e x p re s s io n s f o r t^^ and t ^ a re o b ta in e d
(z = o) = «•* dh ( w , a ) / 6(jj
(z = a) X a ô p /d ü ) -
0 c ( w , a ) / '0 c u
and th u s
"^R “
“ (2 wA /3 o
0
(sinh(aa)oosh(Q :a)(a^+ j(5^)(a^e^+ j3^)
o(ha^f3^ c o s h ^ (a a ) +
When
= d P-j
-
aa/3^(j8^-a^)
)o
s i n h ^ ( a a ) , ** (3o5<,6)
A = J pg; th e e x p re s s io n s f o r th e r e f l e c t i o n
and tr a n s m is s io n i n t e r a c t i o n tim e s a re :
^(sixi(l3^a)oos(l3 ^a )(^ ^- i3p( l3^
+ /3 ^
'(^ § ^ 1
(/îg a ))" ''
(^ 2 ^ )
+
(^ 2
+
. , . . .
(3 ,5 ,7 )
These e x p r e s s io n s c a n n o t he re g a rd e d a lo n e h u t m ust he exam ined
a s p a r t o f th e i n t e g r a l d e s c r ib in g th e r e f l e c t e d o r t r a n s m i t t e d
s ig n a l.
I t m ust he s t r e s s e d t h a t i f
i n th e e x p r e s s io n s ,
th e tim e s T^ and T^ d e riv e d a re n o t th e r e f l e c t i o n and tr a n s m is s io n
tim e s o f th e p u ls e o r s i g n a l as a w h o le.
The e x p r e s s io n s m e re ly
i n d i c a t e t h a t c o n s t i t u e n t w aves, w i t h in a sm a ll ran g e o f f r e q u e n c ie s
aro u n d w w i l l i n t e r f e r e c o n s t r u c t i v e l y and make th e m ost c o n t r i b u t i o n
to th e i n t e g r a l a t tim e tj^ a t p o s i t i o n z = o and a t tim e t ^ a t
p o s itio n z = a.
F o r th e i n c i d e n t p u ls e , th e c e n t r a l o r c a r r i e r fre q u e n c y i s
w^.
As p o in te d o u t i n s e c t i o n ( 1 . 3 ) , i f th e shape o f th e w av ep ack et
i s n e g l i g i b l y a l t e r e d , t h a t i s , th e a m p litu d e f u n c t i o n i n th e
fre q u e n c y dom ain does n o t a l t e r a p p re c ia b ly and th e fre q u e n c y s p re a d
ab o u t
i s n o t l a r g e , t h e n th e s t a t i o n a r y p h ase m ethod y i e l d s a
group t r a v e l tim e and a g ro u p v e l o c i t y .
If
o i s th e p o s i t i o n
-9 3 -
( 3 .5 )
o f th e c e n tr e o f th e w avepacket a t t = o and z i s a su b se q u e n t
p o s i t i o n a t tim e , t , th e n z / t
= v = ôio/3j3 • The n arro w
ê
' S
ë
ra n g e s o f f r e q u e n c ie s im p lie s a p u ls e o f lo n g d u r a tio n i n th e
tim e dom ain,
3 .6
P r e d i c t i o n o f group r e f l e o t i o n and tr a n s m is s io n i n t e r a c t i o n
tipiGS u n d e r c e r t a i n c irc u m s ta n c e s „
I n th e f o llo w in g d i s c u s s i o n , i t w i l l be assum ed t h a t th e ra n g e
o f f r e q u e n c ie s
o f th e image f u n c t i o n o f th e i n i t i a l tim e c o n d itio n
( a G-aussian m o d u la tio n e n v e lo p e i n p a r t i c u l a r ) , ab o u t th e c a r r i e r
fre q u e n c y i s s u i t a b l y c u r t a i l e d .
when
= jp
and
Of p rim a ry i n t e r e s t , i s th e c a se
= a .
The e x p r e s s io n f o r th e m odulus tr a n s m is s io n c o e f f i c i e n t i s
|C /a |
-
2 aj3
(3 * 6 .1 )
o ( ( 05^ ~ jS^)^ s in h ^ ( a a ) + 4
and when a a «
1,
| c/
a
I ^
1.
cosh^ ( # a ) )
Prom e q u a tio n (3 * 3 .1 ) th e i n t e g r a l
f o r th e t r a n s m i t t e d s i g n a l becom es
pOO
1
f;^cu~ (jj^) exp ( j ( w t - (3z)+ c( 0Ü , a ) ) ) ,d (0
( 1 / 2 ti )2
......( 3 * 6 .2 )
-CO
where c(u),a) i s g iv e n i n ( 3 * 3 .4 ) , t h a t i s , f o r
I t h a s b e e n assum ed t h a t th e
w e ig h tin g f u n c t i o n , f (c - w ^ ), h a s n e g l i g i b l e a m p litu d e o u ts id e t h a t
ra n g e .
T hus, f o r a v e ry s h o r t sandw ich s e c t i o n , th e c o n d itio n s o f a
sm a ll fre q u e n c y sp re a d a b o u t t h e c a r r i e r fre q u e n c y and n e g l i g i b l e
change i n th e a m p litu d e f u n c t i o n can be s a t i s f i e d .
A d ap tin g a
d e r i v a t i o n by Bohm (1 1 , C h a p te r 3) o f th e g roup tr a n s m is s io n tim e
f o r a f r e e p a r t i c l e to t h i s w aveguide s e t - u p , i t w i l l be shovm. t h a t
•-94( 3 .6 )
th e t r a n s m i t t e d p u ls e a t 2 = a i s c e n tr e d around t ^ + t ^ where
tgi i s th e tr a n s m is s io n i n t e r a c t i o n tim e c o rre sp o n d in g to th e
c a r r i e r fre q u e n c y
Tfhen 2 = a , th e im a g in a iy ex p o n en t o f
( 3 *6 , 2) i s (w t + 0 ^ ( w , a ) ) where c( w ,a ) = p a + e ^ ( w ,a ) ,
The f u n c t i o n
(w ,a ) may be expanded i n a pow er s e r i e s ab o u t
te r m in a tin g a t th e second pow er.
T h u s,
C^(cü,a)
(w^f 3<) +
(C^ (o) a ) )
(a)-cü^) +
(Ü =: CO
o
*
^
d ^ i C^((0, a ) )
Î
“ ^ 7
(w-Wq)^
—
(0= 0)
o
F o r sm all « a
d (C.| (coya ))
3co
“ “ ^ a e . { ( a ^ + p ^ ) ( a % ^ + § ^ ) - /3^ (/9^ - a ^ ) )
. (4 a ^
c
,
c f)-^
^
2 (>’ %
OOOOOOOO (3*6 o3)
The im a g in a iy e x p o n en t becomes
((ot +CC(o)^^a) 4-
((o^j^a) (co—
(o^ )
h*-
•f Cy («o»a)(co-<o^)^)
and w ith th e s u b s t i t i o n Vf = w-u)^ i n t e g r a l ( 3*6 , 2) i s
( a , t ) = exp (-jco^t
p
+ jC^ (w^, a ) ) .
00
f(W) exp (j(W t + Cjj (o) >a) W + C f (co , a)W^/2)dW
(3 .6 )
-9 5 -
The f u n c t i o n f ( w - w ^) = f (w) i s th e G-aussian f u n c t i o n
exp (-W ^ / 2 A W^).
The i n t e g r a l becomes
exp (jW (t + Cjj) +
(jC " ~ l/AW ^)/2)odW
w hich on c o m p le tin g th e sq u a re o f th e exponent i s
00
8%P ( ( j c y ( aw)
- l ) / 2 (AW)^),dW.
J
(W + j ( t + q)(AW)V(oC!|' (AW)Z _ 1 ) )
o exp ( ( t + C] ) ^ (A W )% (jC ^ (AW)2 - 1 ) )
U sing th e i n t e g r a l
\co
2
1.
e x p (jaU / 2 ) odl! = exp ( jTr/4)'(Tr/(%)^
V
= (2rr (AW)^/(jG!,'(AW)^ - 1 ) ) ^
(t +
L
(AW)^
((G%(AW)2)2_1)2
exp ("jw t +
(w , a ) )
jC '' ( aw)^ ( t + C f)
2 ( ( G" ( à w 2\
f f2 - 1)
f (W)
=
-9 6 -
( 3 .6 )
The a m p litu d e o f t h i s em erg en t wave i s g iv e n by th e e x p r e s s io n
exp( - ( t + e V
(2 a
( A W )^ / 2
( ( g” ( AW )^)^ - 1) ) .
( A W ) ^ / (c!j ( AW)^ - 1) )
T h is i s s t i l l a G-aussian d i s t r i b u t i o n b u t now c e n tr e d around
t|j, £s — Q*
(co^ ^ a )
=
V ^ o
% )(% +
T h is i s th e same e x p re s s io n a s (3 .5 ,& ) f o r sm a ll
o b ta in e d from th e s t a t i o n a r y p h ase a n a l y s i s .
a a andw n
T hus, i t h a s b e en
shown t h a t i t i s p o s s ib l e to u se th e group t r a v e l tim e d e r iv e d
from th e s t a t i o n a r y p h ase a n a l y s i s to d e s c r ib e th e m otion o f th e
w avepacket o r s ig n a l a s a w h o le, w ith c e r t a i n r e s t r i c t i o n s on th e
form o f th e i n i t i a l \Yavepacket and th e m odulus o f th e t r a n s f e r
f u n c tio n o f th e system | c/
a
| ,
E x p re s s io n (3 .6 ,2 f) may b e re a r r a n g e d to g iv e
(a/V g ) ( 1 +
w here v^ =
(i /
/
~
1 /E ^ - t
2
P c ^ /w e ^ ,
2 /
/ 2
......( 3 .6 .5
The e x p re s s io n
2
) may b e i n v e s t i g a t e d f o r th e c o n f ig u r a tio n s
o f f i g u r e s 2a and 2 b ( l 7 ) ,
F o r th e ste p p e d g u id e c o n f i g u r a ti o n ,
E = 1 and i t re d u c e s to
r
((t»o2/“ o^^ - (W c l/* o )^ )/2 (1 - (< »oiA o)^)
and sin c e
alv/ays g r e a t e r th a n a / v
t h i s i s alw ays p o s i t i v e so t h a t ( 3 .6 .5 ) i s
«
Thus th e w avepacket talces lo n g e r to
t r a v e r s e th e v e iy s h o r t l e n g t h , a , o f c u t - o f f g u id e th a n i t does
to t r a v e l th e same d i s ta n c e i n th e u n m o d ified g u id e .
- 97 -
'For
(e
,
> 1 and
- l)(f
(3.6)
(c^)the
+
e x p re s s io n i s
.)*
w hich i s p o s i t i v e when
> 4
^“ o2 /® o )^ (2 +
> (2 + e j / e ^
and n e g a tiv e ?fhen
or
( t u t < e^)
< (2 + ffj,)/
T hus, th e
(tu t >
1)
w avepacket may ta k e lo n g e r o r s h o r t e r to t r a v e r s e
th e
d i s t a n c e , a , compared w ith th e tim e o f t r a v e l i n th e d i e l e c t r i c
f illô d ^ g u id e »
When a a i s vexy l a r g e ,
| b/
a|
"^1
and th e r e f l e c t i o n i n t e r a c t i o n
tim e i s , from (3 .3 * 6 )
tjj =
20) (
+ P^) A P o q C
+ P^)
. . . ( 3 . 6 . 6)
I n th e same way a s above, i t may be shown t h a t , th e r e f l e c t i o n
i n t e r a c t i o n tim e f o r w=
th e c a r r i e r fre q u e n c y may be u se d to
d e s c r ib e th e r e f l e o t i o n i n t e r a c t i o n tim e o f th e G -aussian p u ls e a s
a w h o le.
I t w i l l t h e r e f o r e be c o n clu d ed from t h i s s e c t i o n t h a t f o r v e ry
s h o r t o r v ezy lo n g g u id e s , th e t r a n s m i t t e d and r e f l e c t e d p u ls e s
r e s p e c t i v e l y a re l i t t l e
d i s t o r t e d on i n t e r a c t i o n w ith th e c u t - o f f
s e c t i o n and th e group tim e s o f i n t e r a c t i o n may b e deduced from th e
s t a t i o n a r y phase r e s u l t s f o r
3*7
w = (u ^ .
Ge n e r a l d is c u s s io n o f i n t e r a c t io n tim e f o r d i s t o r t e d p u l s e s .
An i n t e r e s t i n g p o i n t h a s a r i s e n w ith r e g a r d to th e i n t e r a c t i o n
tim e s f o r th e v e ry lo n g g u id e ,
of a » 1 ,
S ince th e r e f l e o t i o n , t ^ ,
and t r a n s m is s io n , t ^ , i n t e r a c t i o n tim e s a re e q u a l, th e tr a n s m is s io n
i n t e r a c t i o n tim e f o r t h i s lo n g g u id e i s g iv en by th e e x p re s s io n i n
-9 8 -
(3 .7 )
e q u a tio n ( 3 o é ,6 ) , w hich i s in d e p e n d e n t o f th e b a r r i e r t h ic k n e s s , a .
A t f i r s t s i g h t t h i s w ould a p p e a r u n r e a s o n a b le „
H ow ever, when th e
m odulus o f th e f u n c t i o n s (c/A ) and (B/A) do n o t rem a in c o n s ta n t a s
i n th e p re v io u s s e c t i o n , i n t e g r a l s ( 3 .5 ,2 ) and (3 .5 * 3 ) do n o t re d u c e
to ( 3 . 6 . 2) .
The m odulus o f th e in te g r a n d
. f (w - w ^) v f ill no
lo n g e r be c e n tre d on th e c a r r i e r fre q u e n c y ,u )^ , b u t on a n o th e r f r e q u e n c y ,
w \
The o u tp u t p u ls e o r w avepacket w i l l be d i s t o r t e d and i t i s no
lo n g e r p o s s ib le to fo llo w th e movement o f th e c e n tr e o r
p eak o f th e
p u ls e u s in g th e r e l a t i o n f o r t ^ , t ^ i n (3 .5 - 6 ) when w = 0; ^ .
I n p r i n c i p l e i t i s p o s s ib l e to f i n d th e fre q u e n c y
!
c o rre s p o n d in g to th e maximum a m p litu d e o f th e d i s t r i b u t i o n
c/A I . f(oJ*-
)»
A f i r s t a p p ro x im a tio n to th e tr a n s m is s io n
i n t e r a c t i o n tim e o f th e d i s t o r t e d p u ls e co u ld be o b ta in e d by
s u b s t i t u t i n g t h i s new v a lu e o f
i n th e e x p re s s io n f o r t ^ . T h u s,
t
s in c e
i s a f u n c t i o n o f th e l e n g t h , a , so a dependence on t h i s
q u a n ti t y h a s b e e n
in tr o d u c e d i n t o th e e x p re s s io n f o r th e t r a n s m is s io n
i n t e r a c t i o n tim e .
However, i t m ust be s t r e s s e d t h a t t h i s e x p re s s io n
i s a f i r s t a p p ro x im a tio n o n ly and h a s b een p r e s e n te d so a s to remove
th e a p p a re n t p a ra d o x c i t e d a t th e b e g in n in g o f t h i s s e c t i o n .
The
tr a n s m is s io n tim e o f th e d i s t o r t e d p u ls e r e q u i r e s e v a lu a tio n o f th e
o r ig in a l in te g r a l in ( 3 .5 .1 ) .
3 -8
P r a c t i c a l a p p l i c a t i o n o f th e a n a l y s i s .
I n s e c t i o n s ( 3 . 6) and ( 3 . 7 ) ^ th e d is c u s s io n h a s b e e n c o n fin e d
to a s tu d y o f th e m o tio n o f th e p e a k o f th e o u tp u t p u l s e , w hich was
ta k e n to d e s c r ib e th e m o tio n o f th e w avepacket a s a w h o le.
In a
co m p lete d i s c u s s i o n , th e v a lu e s o f th e i n t e g r a l s (3 .5 * 1 ) and (3*5*2)
m ust be known b u t t h i s i s n o t p o s s i b l e a n a l y t i c a l l y H o w e v e r , i t
-9 9 -
(3 .8 )
was d e m o n stra te d t h a t f o r a G a u ss ia n a m p litu d e m o d u lated in p u t
p u ls e g th e o u tp u t p u ls e t r a n s m i t t e d th ro u g h a v e ry s h o r t s e c t i o n
o r r e f l e c t e d from a v e ry lo n g c u t - o f f s e c tio n was n e g l i g i b l y
d is to rte d .
The i n t e r a c t i o n tim e , a s deduced from th e s t a t i o n a i y
p h ase m ethod, f o r w =
, c o u ld be u se d to d e s c r ib e th e m o tio n o f
p u ls e as a w hole as shown by th e s h i f t o f th e p e ak .
The r e s t r i c t i o n s
o f th e a n a l y s i s a r e th e s e :
I
The m odulus o f th e t r a n s f e r f u n c t i o n ,
1b/
a
| o r | c/ a |
o f th e c u t - o f f s e c t i o n i s c o n s ta n t o r changes v e ry l i t t l e o v e r th e
ran g e o f f r e q u e n c ie s c o n s id e re d .
II
The ran g e o f f r e q u e n c ie s a b o u t th e c a r r i e r fre q u e n c y o f th e
i n i t i a l p u ls e i s n a rro w and c o n ta in e d v /ith in th e ra n g e
< w <
w 2 o r above
W h ils t a G a u ssia n a m p litu d e m o d u lated in p u t w avepacket o r
p u ls e h a s b een u se d i n a c c o rd a n c e w ith th e quantum m e c h a n ic a l
a p p ro a ch , th e G a u ssia n f u n c t i o n c o u ld be r e p la c e d by a s i m i l a r
am p litu d e m o d u la tin g f u n c t i o n , w ith a s i m i l a r im age f u n c tio n i n
th e fre q u e n c y dom ain, c e n tr e d a ro u n d
and c u r t a i l e d to a n a rro w
ran g e ab o u t t h i s c a r r i e r f re q u e n c y .
I n C h a p te r A? a p r a c t i c a l m icrowave system i s d e s c r ib e d f o r
i n v e s t i g a t i n g th e i n t e r a c t i o n o f c a r r i e r p u ls e s w ith s e c t i o n s o f
c u t - o f f g u id e sandw iched betw een two i d e n t i c a l p ro p a g a tin g g u id e s
( f i g u r e s 2 a, 2l()17))The d e t e c t i n g system allo v fs m easurem ent o f th e
tim e d is p la c e m e n t o f th e p eak o f th e p u ls e o n ly .
Thus th e d is c u s s io n
i n s e c t i o n ( 3 .6 ) w ith c o n d itio n s I and I I can be r e l e v a n t to th e
p r a c t i c a l i n v e s t i g a t i o n u n d e rta lc e n ,
F o r th e v e ry s h o r t g u id e , tr a n s m is s io n i n t e r a c t i o n tim e s w ere
p r e d i c te d , from e x p re s s io n ( 3 . 6 .A) f o r th e p r a c t i c a l sy stem ^
-1 0 0 ( 3 .8 )
C o n sid e r f o r i n s t a n c e , a c a r r i e r fre q u e n c y o f 6*02 G-Kla and fo ^ =
4.3G-HZ (wo. 14) .
70Hz (sy /X
!For s h o r t s e c tio n s o f g u id e w ith c u t - o f f f r e q u e n c ie s
= . 023) and 7 .5 OHz ( a /X ^
= .023) th e p r e d i c te d
tr a n s m is s io n i n t e r a c t i o n tim e s a re .0 1 8 and ,011 n se o s r e s p e c t i v e l y .
I n tim e o f t r a v e l m easurem ents w ith u n d i s t o r te d p u l s e s , i t was
fo u n d p o s s ib le t o m easure sub n a n o sec o n d tim e s w ith an e r r o r o f
,01 n se o s ( s e e f i g u r e s 27a( 121 ) and27b(l22)).
T h e re fo re th e
p r e d i c t e d tim e s above d id n o t come w ith in th e r e s o l u t i o n o f th e
m easu rin g system .
T h is , th e n , w i l l n o t form p a r t o f th e p r a c t i c a l
in v e s t i g a t i o n ,
F o r th e v e iy lo n g c u t - o f f s e c t i o n , c o n d itio n s I and I I can be
s a t i s f i e d f o r th e r e f l e c t e d p u ls e .
F ig u re s 3 7 ( 140) ahd 39('l42) a r e
g ra p h s o f th e modulus o f th e r e f l e c t i o n c o e f f i c i e n t
th e
fre q u e n c y , f .
|b / a |
v e rsu s
F o r a g iv e n c u t - o f f fre q u e n c y , th e c u rv e s
c o rre s p o n d to t h r e e d i f f e r e n t l e n g t h s , a , o f i n te r p o s e d s e c t i o n .
I t can be se e n t h a t , f o r th e v e ry lo n g g u id e o f l e n g t h s 30oms, th e
r e f le c tio n c o e f f ic ie n t |b / a |
c u t-o ff.
i s a lm o st c o n s ta n t and u n i t y
below
W ith th e 1,6cm l e n g t h , t h e r e i s a w ide v a r i a t i o n from th e
lo w e r to th e h ig h e r c u t - o f f fre q u e n c y .
F ig u r e s 3Q(141) an d 40(143)
show th e v a r i a t i o n o f r e f l e c t i o n i n t e r a c t i o n tim e w ith th e fre q u e n c y ,
f ,
S in ce th e tim e s a r e w ith in th e r e s o l u t i o n o f th e m easu rin g
sy ste m , 30cm lo n g " c u t - o f f " s e c t i o n s w ere d e sig n e d so t h a t t h e i r
t h e o r e t i c a l l y p r e d i c t e d tim e s o f i n t e r a c t i o n were i n th e p o s s ib l e
m ea su rin g ra n g e .
F o r d i s t o r t e d o u tp u t p u l s e s , th e a n a l y s i s can be c a r r i e d o u t
and i n t e r a c t i o n tim e s p r e d i c t e d o n ly i f th e fre q u e n c y f u n c t i o n o f
th e in p u t p u ls e i s c o m p le te ly Imown,
T h is , o f c o u rs e , im p lie s a
com plete know ledge o f th e a m p litu d e d i s t r i b u t i o n o f th e i n p u t p u l s e .
-1 0 1 -
( 3 .8 )
The p r e s e n t m e a su rin g system d id n o t g iv e t h i s com plete in f o r m a tio n
and so i t was r e a l i z e d b e fo r e any tr a n s m is s io n e x p e rim e n ts w ith
c u t - o f f s e c t i o n s o f medium l e n g t h w ere a tte m p te d , t h a t / co m p ariso n
w ith t h e o r e t i c a l l y p r e d i c t e d r e s u l t s would n o t be p o s s i b l e .
To sum m arize, th e r e f l e c t i o n i n t e r a c t i o n o f c a r r i e r p u ls e s
w ith lo n g c u t - o f f g u id e s w i l l be i n v e s t i g a t e d p r a c t i c a l l y i n an
a tte m p t to compare th e i n t e r a c t i o n tim e s w ith th o s e p r e d i c t e d
t h e o r e t i c a l l y i n e q u a tio n ( 3 . 6 . 6) .
I t w i l l be assum ed t h a t th e
movement o f th e p eak o f t h e p u ls e d e s c r ib e s th e movement o f th e
p u ls e as a w h o le.
However, th e e x p e rim e n ts vd-ll n o t be c o n fin e d to th e s e b u t
w i l l be e x te n d e d to t r a n s m is s io n e x p e rim e n ts th ro u g h medium le n g th
s e c t i o n s o f g u id e whose c u t - o f f f r e q u e n c ie s may be above o r below
th e c a r r i e r fre q u e n c y .
F o r th e n o n - c u to f f i n te r p o s e d s e c t i o n ,
e x p re s s io n (3 * 5 .7 ) may be u se d f o r r e f l e c t i o n / t r a n s m i s s i o n i n t e r ­
a c t i o n tim e s and e q u a tio n s (3 .4 * 1 ) and ( 3 .4 .2 ) s u i t a b l y amended
fo r
3*9
= 0 P-)
Y2= Ü Pg-
F o o tn o te on th e quantum m ec h n io a l tu n n e l e f f e c t .
I t V7as m en tio n ed p r e v i o u s ly t h a t i n t e r e s t c e n tr e d on t h i s
tu n n e l e f f e c t i n c o n n e c tio n w ith i t s p o s s ib l e c o n t r i b u t i o n to th e
l i m i t a t i o n o f th e fre q u e n c y re s p o n s e o f d e v ic e s b a s e d on th e e f f e c t ,
( 35^ 1967)
Hov/ever, t h e r e h a s b e e n some r e c e n t d is c u s s io n by T h o rih :er e t a l^
^
^
i n which th e p h y s ic a l s i g n i f i c a n c e o f tu n n e lin g tim e h a s b e e n
q u e s tio n e d .
They c o n s id e r th e tr e a tm e n t d e s c r ib e d above a s th e
" q u a s i c l a s s i c a l " o n e, g iv in g a " tr a n s m is s io n tim e " .
A " tra n s itio n
tim e " o f an e l e c t r o n tu n n e lin g from a s t a t e on one s id e o f th e
b a r r i e r to a s t a t e on th e o t h e r s id e can a ls o be d e te rm in e d .
" t r a n s i t i o n " and " tr a n s m is s io n " tim e s a re q u ite d i s t i n c t *
The
I n th e
( 3 . 9)
-102
SO c a l l e d " q u a s i - c l a s s i c a l ” t r e a tm e n t, th e r e f l e c t i o n and
tr a n s m is s io n tim e s o f i n t e r a c t i o n o f a w e ll d e fin e d e l e c t r o n p a c k e t
w ith th e h a r r i e r can be c a l c u l a t e d .
To use th e s e tim e s f o r th e m e ta l-
i n s u l a t o r - m e ta l tu n n e lin g d e v ic e , i t i s n e c e s s a r y to s p e c if y w hich
e l e c t r o n v / i l l t u n n e l o r w hich e l e c t r o n h a s tu n n e le d o r t h a t i t
t u n n e ls on one e n c o u n te r w ith th e b a r r i e r .
The t r a n s i t i o n tim e i s
v e ry d i f f e r e n t from th e t r a n s m i s s io n tim e .
T h o rn b e r co n clu d es t h a t
i t i s th e t r a n s i t i o n tim e w hich i s th e p h y s i c a l l y m e a n in g fu l and
c h a r a c t e r i s t i c tim e o f m e t a l - i n s u l a t o r - m e t a l d e v ic e s .
How ever, t h i s
does n o t d e t r a c t from th e i n t e r e s t i n g m odel o f th e quantum m e c h a n ic a l
tu n n e lin g o f an e l e c t r o n on an e n c o u n te r w ith th e b a r r i e r .
The
t u n n e lin g c u r r e n t d e te rm in e d by b o th a p p ro a c h e s a re i d e n t i c a l .
-*-l03“-
(4 .0 )
CH&FTm k.
EXPERIMENTAL IMVESTICrATIONS ïfITH NANOSECOND MICROWAVE CARRIER HJLSES
4 .0
I n trI’lomdli' uI*-iror, tio ia
The f i r s t t e n t a t i v e e x p e rim e n ts to i n v e s t i g a t e th e i n t e r a c t i o n
o f a nanosecond microwêtve c a r r i e r p u ls e v /ith a s e c t i o n o f w aveguide
c u t - o f f w ith r e s p e c t to th e c a r r i e r fre q u e n c y a r e d e s c r ib e d b r i e f l y *
Hov/ever, th e m a jo r p a r t o f th e d e s c r i p t i o n i s g iv e n to th e e x p e rim e n ta l
r e a l i z a t i o n o f th e system i n v e s t i g a t e d t h e o r e t i c a l l y i n th e p r e v io u s
c h a p te r .
The r e s u l t s a re c a r e f u l l y e v a lu a te d and com pared m t h th e
t h e o r e t i c a l l y p r e d i c t e d o n es where f e a s i b l e .
d iv id e d i n t o th r e e p a r t s .
The c h a p t e r i t s e l f i s
P a r t I i s co n cern ed v d th th e p r o d u c tio n
o f th e nanosecond c a r r i e r p u ls e s and th e i n i t i a l e x p e rim e n ts .
P a rts
I I and I I I d e s c r ib e e x p e rim e n ts w ith th e w aveguide arfang© m ants o f
f i g u r e s 2a and 2b r e s p e c t i v e l y ,
4.1
G e n e ra tio n and jm e a s u re i^ n t te c h n iq u e s o f nano seco n d
les
The g e n e r a tio n and m easurem ent te c h n iq u e s o f nanosecond
m icrow ave p u ls e s h a s b een d e s c r ib e d i n d e t a i l by I t o ^ ^ ^ \
The g a te d
t r a v e l l i n g wave tu b e i s amongst s e v e r a l m od u lated c o n tin u o u s wave
m ethods m en tio n ed .
Beck and M andeville^^*^^ and D o g a d k i n ^ h a v e
u se d t h i s m ethod i n i n v e s t i g a t i o n s i n th e 3 cm, b a n d .
i l l u s t r a t e s th e m ethod o f m o d u la tio n .
F ig u re 19
The p u ls in g o f th e beam
fo rm in g e le c t r o d e from a ”c u t - o f f " c o n d itio n to norm al o p e r a tin g
c o n d itio n s f o r a v e ry s h o r t tim e p ro d u c e s a s h o r t p u ls e o f o u tp u t
en erg y from th e c,w , i n p u t .
The " c u t - o f f " c o n d itio n o f th e
)
•1
(4»1)
Figure 19o
V
o u tp u t
of
t,w ,t.
( o .w .)
—t im e i r i ~ ^
nseo s.
CGtim e ’ in
nseo s,
F ig u re 19%
F ig u r e 1 9 .
D iagram i l l u s t r a t i n g th e p ro d u o tio n
o f n an o seco n d m icrow ave c a r r i e r p u l s e s
by v a r i a t i o n o f th e c o n tr o l g r i d p o t e n t i a l ,
w ith r e s p e c t to t h e cathode ( a f t e r
D ogadM n^ 38)^^
j
-1 0 5 -
( 4 .1 )
t r a v e l l i n g viave tu b e i s p ro d u ce d by a l a r g e n e g a tiv e b i a s ,
The l a r g e
p o s i t i v e i n p u t v id e o p u ls e r e t u r n s th e t . u . t , to i t s r a t e d d .o ,
o p e r a tin g v a lu e ( f i g u r e 19b) and th e o u tp u t microwave p u ls e i s
p ro d u ce d ( f i g u r e 1 9 c ).
As I t o
( i n 1965) p o i n ts o u t , t h e r e does n o t e x i s t a t p r e s e n t
c o m m erc ially a v a i l a b l e means f o r o b s e rv in g a s i n g l e c a r r i e r p u ls e
d i r e c t l y v /ith p u ls e c a r r i e r f r e q u e n c ie s much above 2,5GHz.
The p r i n c i p a l a d v a n ta g e o f h e te ro d y n e d e m o d u la tio n , i n w hich
th e c a r r i e r p u ls e i s m ixed w ith a l o c a l s i g n a l much s t r o n g e r th a n th e
p u ls e pealc s i g n a l , i s t h a t i t y i e l d s in f o r m a tio n o f b o th th e en v elo p e
o r a m p litu d e m o d u la tio n and th e p h a se m o d u la tio n .
A ls o , Imowledge
o f th e c h a r a c t e r i s t i c s o f th e microv/ave dio d e m ix e r i s n o t e s s e n t i a l .
T h is i s n o t so when a c r y s t a l r e c t i f i e r , u se d i n c o n ju n c tio n v /ith an
U.C. lo a d i s u se d t o r e c o v e r th e e n v e lo p e o f t h e c a r r i e r p u l s e ,
g ( t ) = x ( t ) cos (w ^t +
T h is i s p o s s i b l e when t h e r e
9 ^ ( t ))
i s n e g l i g i b l e o v e rla p b etw een th e s p e c t r a
o f th e e n v elo p e f u n c t i o n and th e c a r r i e r p u l s e .
re s p o n se to n an o seco n d c a r r i e r
a tte n u a tio n c h a r a c te r is tic s
The o v e r a l l d e t e c t o r
p u ls e s depends on
th e p h ase and
o f th e in p u t c i r c u i t
and o u tp u t c i r c u i t
and th e b e h a v io u r o f th e c i y s t a l c u r r e n t s e n s i t i v i t y w ith fre q u e n c y .
l% iitfo rd ^
h a s made a n e x p e rim e n ta l in v e s t i g a t i o n o f th e fre q u e n c y
re s p o n s e o f n an o seco n d c a r r i e r p u ls e d e t e c t o r s .
In a d e q u a te b an d w id th
o f th e in p u t and o u tp u t v id e o c i r c u i t s w i l l g iv e e n v e lo p e d i s t o r t i o n ,
a s T /ill any d e p a rtu re from th e q u a d r a tic c h a r a c t e r i s t i c s o f th e d io d e,
A, 2
P re v io u s p u ls e i n v e s t i g a t i o n s i n w aveguide
T hese i n v e s t i g a t i o n s f a l l i n to two c a t e g o r i e s .
The f i r s t one
in c l u d e s e x p e rim e n ts i n w hich in f o r m a tio n on b o th th e a m p litu d e and
- 106p h a se m o d u la tio n o f th e p u ls e i s loio'iTn,
(4 . 2 )
I n I t o ' s e x p e rim e n ts
in
p a rtic i^ L a r, th e d i s p e r s i o n o f a G-aussian a m p litu d e m od u lated p u ls e a s
i t t r a v e l l e d a l a r g e d is ta n c e ('^ lO O c u t - o f f w a v e le n g th s) i n a
w aveguide v/as m easured and fo u n d to be i n good ag reem en t w ith
t h e o r e tic a lly p r e d ic te d re s u lts *
The seco n d c a te g o ry i n c l u d e s e x p e rim e n ts i n vdiich th e e n v elo p e
o n ly i s d e te c te d i n tim e o f t r a v e l m easurem ents, f o r e x a m p l e ^ »(41 ) «
To th e a u t h o r ’ s know ledge, t h e r e e x i s t no p r e v io u s i n v e s t i g a t i o n s
o f th e i n t e r a c t i o n o f m icrowave c a r r i e r p u ls e s w ith s e c t i o n s o f g u id e
c u t - o f f w ith r e s p e c t to th e c a r r i e r fre q u e n c y ,
4 .3
The a i m o f th e e x p e rim e n ts
The b a s i c aim o f t h i s i n v e s t i g a t i o n h as b e e n to malce a tim e
d ep endent s tu d y o f th e i n t e r a c t i o n o f e le c tr o - m a g n e tic waves w ith a
s e c t i o n o f " c u t - o f f " g u id e .
T h is h a s n e c e s s a r i l y in v o lv e d u s in g
m icrowave c a r r i e r p u ls e s ( s e e s e c t i o n (1 .3 ) ) «
The i n i t i a l t e n t a t i v e
e x p e rim e n ts in v o lv e d p a s s in g m icrowave p u ls e s th ro u g h a t e s t s e c t i o n
c o n s i s t in g o f i n p u t and o u tp u t c o u p lin g lo o p s i n a s h o r t l e n g th o f
" c u t - o f f " c i r c u l a r g u id e , shown i n f i g u r e 20.
R e c o rd in g s o f th e
o u tp u t p u ls e s f o r v a r io u s l e n g t h s o f guide b etw een th e in p u t and
o u tp u t lo o p s a re shown i n f i g u r e 21 (1C6 )
f ^ =6,275G-Hz*
f o r a c a r r i e r fr e q u e n c y ,
I t d id n o t p ro v e p o s s ib le to i n t e r p r e t th e s e r e s u l t s
o r deduce tim e d e la y s f o r th e t r a n s m i t t e d p u l s e ;
n e i t h e r c o u ld th e
sy stem be d e s c r ib e d sim p ly i n m a th e m a tic a l te rm s .
However, i t i s p o s s ib l e to d e s c r ib e th e system o f f i g u r e s
2 a , 2b ( ] 7 ) and t h i s h a s b e en done i n C h a p te r 3»
T h e re , th e a n a lo g y
h a s been draw n betw een th e tim e d ep en d en t i n t e r a c t i o n o f e l e c t r o n
wave p a c k e t w ith a p o t e n t i a l b a r r i e r , th e so c a l l e d quantum m e c h a n ic a l
■•107/|\
4
(4 .3 )
o u tp u t to
sa m p lin g
o s c illo s c o p e
n y lo n s u p p o rt
sp rx n g
fin g e rs
\
EigicL coa3d.al
lin e
B ra ss
s p r in g
fin g e rs
c o u p lin g ■
lo o p
N ylon s u p p o rt
w ith e n g ra v e d
s c a le .
B ra s s
p la te
Type *N*
p lu g
i n p u t fro m t . w . t .
2t
R ig id
c o a x ia l
lin e
F ig u r e 20c. L au n ch in g
o r p ic k -u p h e a d
F ig u r e 20a) C i r c u l a r g u id e
c o n f i g u r a ti o n
p la n
- s id e view
r = 1 .3 5 oms,
F ig u r e 2 0 .
fo r
i
1 .5 5 o m s/
S p rin g f i n g e r
s u p p o rts ,
s u rro u n d in g th e
c o u p lin g lo o p s .
R ig id c o a x ia l
lin e .
/N
(fi—
F ig u r e 20h. L au n ch in g
o r pick«"Up h e a d - f r o n t
view
C o a x ia l c z y s t a l
d io d e
uX
,1
mode = 3,41 r
The c i r c u l a r g u id e c o n f i g u r â t io n
i n p r e lim in a r y m ea su re m e n ts.
u se d
-1 0 8 -
(V.3)
'Ê
I
f
f
I — >tim e
1cm s
.9 5
n seos.
a m p litu d e
C a r r ie r fr e q u e n c y o f in c id e n t p u ls e
C u t-o ff fr e q u e n c y o f
m ode
= 6 . 2 7 5 GHz
= 6.52G rH z
R e c o r d i n g **oc” w a s m ad e w i t h t h e s p r i n g f i n g e r s
o f th e la u n c h in g and p ic k -u p h e a d s to u c h in g .
The s e p a r a t io n o f th e la u n c h in g and p ic k
l o o p s w a s i n c r e a s e d i n s t e p s o f 2mm.
up
F ig u r e
tr a n s m itte d
21.
R e c tifie d
th r o u g h
e n v e lo p e
th e
o f fig u r e
o f p u ls e s
c i r c u l a r g u id e
20.
c o n fig u r a tio n
-1 0 9 -
( 4 .3 )
tu n n e l e f f e c t and th e i n t e r a c t i o n o f a G-aussian a m p litu d e m o d u la ted
c a r r i e r p u ls e w ith a s e c t i o n o f g u id e , whose c u t o f f fre q u e n c y i s
g r e a t e r th a n th e c a r r i e r fre q u e n c y o f th e p u l s e .
I t was shown t h a t
u n d e r c e r t a i n c o n d itio n ^ , t h i s tim e d ependent a n a l y s i s y i e l d e d
e x p re s s io n s f o r r e f l e c t i o n and tr a n s m is s io n i n t e r a c t i o n tim e s f o r
t h i s p u ls e ( s e e s e c t i o n ( 3 . 8 ) ) .
v a lu e s o f f ^ ^ , f ^ g ,
f^ and a
From th e s e e x p r e s s io n s , f o r g iv e n
i t was fo u n d p o s s ib l e to p r e d i c t
r e f l e c t i o n i n t e r a c t i o n tim e s vdiich w ere w i t h in th e re s o l- v in g pow er
o f th e d e te c t i n g system u se d f o r th e i n i t i a l e x p e rim e n ts .
T h u s, i t
a p p e a re d p o s s ib l e to u se th e e x p re s s io n s to d e s ig n s u i t a b l e t e s t
" c u t-o ff" s e c tio n s .
To sum m arize b r i e f l y , th e aim o f th e s e e x p e rim e n ts i s to
r e a l i z e p r a c t i c a l l y th e sy stem d e s c r ib e d t h e o r e t i c a l l y i n C h a p te r
3 and to u se th e r e s u l t s o f i t s a n a l y s i s to d e s ig n s u i t a b l e t e s t
s e c t i o n s o f th e form i l l u s t r a t e d i n f i g u r e s 2 a , 2b ( 1 7 ) ,
Where
f e a s i b l e , tim in g m easurem ents on th e s e t e s t s e c t i o n s w i l l be
compared w ith th e t h e o r e t i c a l l y p r e d i c t e d v a lu e s .
T h is tim e
d ep en d en t work w i l l form a n a t u r a l e x te n s io n o f th e c.w . m icrow ave
a n a lo g y e x p e rim e n ts o f th e quantum m ech an ica l t u n n e l e f f e c t o f
(7 ) and ( 8 ) .
if,if.
P ro d u c tio n o f th e n an o seco n d m icrowave p u ls e s
The ra n g e o f c a r r i e r f r e q u e n c ie s o f th e s e n an o seco n d p u ls e s was
d e te rm in e d by t h a t o f th e t . w . t . w hich was u se d f o r t h e i r p ro d u c tio n *
T h is was a M u lla rd LB6/10 w ith fre q u e n c y ran g e 3 * 9 5 to 6,ifG-Hz and
g a in o f a b o u t 34âB f o r o u tp u t pow er o f 3 w a t t s .
tu b e was ch o sen f o r i t s h ig h g a in and S a t u r a t i o n
T h is p a r t i c u l a r
o u tp u t pow er
(10 w a tts ) i n a n t i c i p a t i o n o f t h e s i g n a l a t t e n u a t i o n on t r a n s m is s io n
**“i 10*^
th ro u g h c u t - o f f s e c t i o n s .
(^-•4*)
I t s t y p i c a l worlcing p o t e n t i a l
d i f f e r e n c e s and c u r r e n t s u n d e r n orm al o p e r a tin g c o n d itio n s a re
shovm i n f i g u r e 22b(111 ) .
D uring p u ls in g c o n d it i o n s , th e c o n tr o l
g r i d o f th e tu b e , u s u a l ly h e ld a t 8 v o l t s n e g a tiv e w ith r e s p e c t
to th e c a th o d e , was h e ld a t 340 v o l t s n e g a tiv e .
c u t - o f f th e tu b e .
T h is e f f e c t i v e l y
The tu b e was th e n sw itc h ed on by a p p ly in g a
p u ls e to t h i s g r i d o f 2l\0 v o l t s p o s i t i v e .
A T e k tro n ix p u ls e
g e n e r a to r ty p e 109 w ith an e x t e r n a l 600 v o l t s s u p p ly was u se d f o r
t h i s p u rp o se .
F ig u re 2 3 a ( ll2 ) i l l u s t r a t e s th e p u ls e from th e
p u ls e g e n e r a t o r .
F ig u re 22a(l11 ) i s a b lo c k d iag ram o f th e a p p a r a tu s
u se d f o r th e p r o d u c tio n o f n a n o sec o n d microwave p u l s e s .
I n th e i n i t i a l s e t t i n g up o f th e a p p a r a tu s , th e t . w . t . was
c o r r e c t l y a li g n e d u n d e r norm al c .w . c o n d itio n s .
v o lta g e was th e n a d ju s te d f o r p u ls e wo riding.
The g r i d b i a s
I t was found
n e c e s s a r y to a d j u s t b o th th e c o n tin u o u s wave in p u t and th e h e l i x
v o lta g e to o b t a i n a s u i t a b l e o u tp u t p u ls e .
A p u ls e was c o n s id e r e d
s u i t a b l e when th e o s c illo g r a m o f th e d e te c te d p u ls e showed a
c l e a r l y d e fin e d , a p p ro x im a te ly sy m m e tric a l o u t l i n e and a z ero
le v e l b a s e lin e .
T h is l a t t e r c o n d itio n e n su re d t h a t any n e g a tiv e
o v e rs h o o t a t th e f a l l i n g edge o f th e p u ls e was n e g l i g i b l e .
F ig u re
23b ( l 12) shows th e r e c t i f i e d e n v elo p e o f a t y p i c a l p u ls e from th e
t.w .t.
E x p e rim e n ts w ere p e rfo rm e d w ith two ty p e s o f t e s t s e c tio n
and th e s e m .11 be d e s c r ib e d i n t h e fo llo v /in g s e c t i o n s .
(4 . 4)
•111
40(3B. c o u p le r f o r
t r i g g e r p u ls e to
sam p lin g o s c i l l o s c o p e
T e k tr o n ix Type 661
w ith 5TIA tim in g
u n it
n s e c . m icrowave
c a r r i e r p u ls e
is o la to r
M u lla rd LB6/10 t . w . t ,
1. C o l le c t o r
2 . H e lix
3 . F o c u sin g m agnet
4 . Anode
5 . G rid
6 . C athode
c.w . m icrow ave
^ in p u t (VfG14)
CCol
T e k tro n ix ty p e 109
n s e c . p u ls e g e n e r a to r
F ig u r e s 2 2 a .
B lo ck d iag ram o f t h e a rra n g e m e n t f o r th e
p r o d u c tio n o f n an o seco n d m icrowave c a r r i e r p u l s e s ,
p o t e n t i a l w ith
r e s p e c t to
c a th o d e
c o lle c to r
1*7 kV, V
h e lix
2 .6 5
anode
1 .9 5 KV,
g rid
F i g u r e 22b.
-8
Vpg
c u rre n t
0 .2 5 mA
VCG
T y p ic a l o p e r a t i o n o f t . w . t .
-112
(4.4)
a m p litu d e
I
'
» tim e
1om s
^ipixjoce 2 3 a ,
R e c o r d in /» o f t h e
p u ls e
fr o m
th e
p u ls e
«93 n seo s
g e n e r a to r
a m p litu d e
ÎI
^ tim e
1cm a
#93 n seo s
a 6 # 0 1 GHz
F i/g u r e
23b#
R e c tifie d
at
e n v e lo p e
o f th e
th e o u tp u t o f th e
p u ls e
t .w .t .
d e te c te d
-1 1 3 PART I I ,
4 .5
(4 .5 )
- The S tep p ed w aveguide T e s t S e c tio n
R e f le c t i o n i n t e r a c t i o n e x p e rim e n t w ith lo n g s 31id.es
T h is t e s t s e c t i o n , i n w hich th e in c r e a s e i n c u t - o f f fre q u e n c y
w ith r e s p e c t to th e m ain guicle v/as achieved, hy re d u c in g th e b ro a d
dim en sio n o f th e g u id e , was u s e d to f i n d th e tim e s o f i n t e r a c t i o n
on r e f l e c t i o n from i t s f r o n t s u r f a c e .
I t c o n s is te d o f w aveguide WG14 p ro p a g a tin g t h e
mode
(f^ ^ = 4 o3G-Hz) and i t can be assum ed t h a t £?ny w aveguide o r w aveguide
com ponents r e f e r r e d to a re o f ty p e lTG-14 u n le s s s p e c i f i c a l l y d e s c r ib e d
a s o th e r w is e .
The o u tp u t p u ls e from th e t , w , t , was f e d i n t o th e H arm o f a
m agic t e e s e c t i o n , a s shown i n f i g u r e 24(114 ) .
la u n c h e d i n to th e m ain arm s, L and R,
The p u ls e was
L c o n ta in e d a m atched l o a d ,
th e r e b y e l i m in a ti n g r e f l e c t i o n s from t h i s arm.
The p u l s e , i n c i d e n t
i n arm R p ro c e e d e d to th e te r m in a t io n a t A and was r e f l e c t e d ,
e n te r in g th e E arm o f th e m agic t e e , whence i t was d e te c te d and
re c o rd e d .
The s te p p e d w aveguide u n i t i t s e l f c o n s is te d o f a 30om l e n g th o f
p l a i n w av eg u id e, w ith t i g h t l y f i t t i n g b r a s s p l a t e s , P , i n s e r t e d
p a r a l l e l to th e n arro w w a ll a s shown i n f i g u r e 2 5 a ( ll5 ) #
p l a t e s were a ls o 30cms lo n g , , 5 oms w ide and 1 ,3 8 cms h ig h .
These
E ach
was s u p p o rte d l a t e r a l l y i n th e g u id e by th r e e p a i r s o f sc re w s.
The
b ro a d d im en sio n c o u ld be f u r t h e r re d u c e d by p l a c in g sm a ll b r a s s
i n s e r t s , I , betw een th e lo n g b r a s s p l a t e s and th e n arro w w a ll:
f i g u r e 23b ,
T hese wore 1 ,3 8 oms h ig h and 1 cm lo n g .
The re d u c e d
b ro a d d im en sio n s a r e l i s t e d i n t h e f i r s t row o f f i g u r e 26(116)
w ith th e G orre& ponding c u t - o f f f r e q u e n c ie s f o r th e
mode i n
.
(4.5)
•114'
in p u t
WG- 14 w aveguide
i n c i d e n t p u ls e
p la in
\AAA
g u id e
i—r e f l e c t e d p u ls e
m atched
lo a d
o u tp u t
W a v e g u id e / c o a x ia l
tr a n s f o r m e r
Sander* s C o a x ia l
C r y s t a l h o l d e r CDIÿ^S
w ith CV215^ c r y s t a l
Sam pling o s c i ll o s c o p e
( T e k tr o n ix 66I* )
X Y R e c o rd e r
(H oneyw ell)
F ig u re 24.
.115-
(4 .5 )
30cms. p l a i n g u id e , WGr14
S
L
»r.,,.r.r-rT-: r , i r i i i i r r-n pj
.
..
rr-srrxixErnc]
F ig u re 2 3 a.
The s te p p e d v /av eg u id e t e s t s e c t i o n - p l a n .
P
\
F ig u r e 25b.
I
The s te p p e d w aveguide t e s t s e c t i o n
WG14
- f r o n t view
- 1 16 -
(4.5 )
----------- 1
Reduced w id th o f
m o d ifie d g u id e
i n oms.
Cut o f f fre q u e n c y
f o r H . mode i n
&H
z
R a tio o f th e
c a r r i e r and c u t­
o f f f r e q u e n c ie s o f
th e m o d ifie d g u id e
R a tio o f th e c u t­
o f f f r e q u e n c ie s
o f th e m o d ifie d
and u n m o d ified
g u id e s
1 .9 3
1 .8 5
6 .9 9
7 .3 4
7 .6 9
8.11
.89
.8 3
.81
.7 7
.7 3
1 .3 6
1 .6 3
1.71
1 .7 9
1 .8 8
2 .3 3
2 ,2 4
6 ,1 2
6 ,4 4
6 .7
.3 1
.9 2
1 .4 2
■ ■ re fle c tio n
i n t e r a c t i o n tim e
n s e c s ,c o r r e s p o n d ­ .3 2
in g to th e c a r r i e r
fre q u e n c y , f .
( p r e d ic te d ) °
R e f le c t i o n i n t e r s
a c t io n d e la y tim e
.4
i n n s e c s , m easured
a t 25cmSo
1 .3
2 .1 4 3
.1 9
.1 3
.125
.1 0 7
,0 9 3
.0 8 3
.2 8
.16
.1 3
.0 8
.0 7
.0 7
.5
.3 2
.2 0
.1 4
.11
.0 7
.0 6 5
.9 7
.9 2
.8 9
.8 5
.8 7
.8 7
.8 5
sh o rt
c i r c u i t .8 7
.8 9
.86
.8 3
.8 7
.8 6
M easured a t 34 oms
M easured
d e la y tim e
i n n se o s
f o r e x tra
18oms p a th
le n g th f o r
re fle c tio n
..from ......
2 .0 4 4
2 .4 5
m o d if­
ie d
g u id e
1
Pigure 2 6 T ab le o f M easurem ents f or re d u c e d w aveguide d im en sio n s
and m easured tim e s f o r f^
itesisA^rsittSîStigvîil
^
-1 1 7 th e second row*
( 4 .5 )
T h is m o d ifie d g u id e was fo llo w e d by a le n g th o f
p l a i n g u id e and a m atched lo a d ?
( l ’o r c o n v e n ie n c e , th e m o d ifie d
g u id e and th e m atched lo a d a re r e f e r r e d to a s th e " s te p p e d
w aveguide u n i t" » )
The frec[uenoy o f th e c o n tin u o u s wave in p u t to
th e p u ls e d t * w ,t . was f i x e d a t a c o n s ta n t v a lu e ( f o r a re a s o n
w hich T /ill be e x p la in e d l a t e r ) i n th e ran g e 3 «95 to
T h is
d e te rm in e d th e c a r r i e r fre q u e n c y o f th e p u ls e and a ls o to some
e x te n t th e shape and le n g th o f p u ls e la u n c h e d i n t o th e lo n g arm o f
th e m agic t e e s e c tio n *
4 «6
E xperim en t a l P^rp c e d p re
T h is c o n s i s t e d o f two p a r t s :
f i r s t l y , f i n d in g th e most
s u i t a b l e p o s i t i o n i n th e lo n g arm o f th e m agic t e e s e c t i o n f o r
talcin g r e c o r d in g s o f p u ls e s r e f l e c t e d from th e t e r m in a t io n , A,
and s e c o n d ly , ta lc in g r e c o r d in g s w ith th e t e s t s e c tio n *
I n th e f i r s t p a r t , s e v e r a l l e n g t h s o f p l a i n g u id e were
trie d .
c irc u it.
Each was te rm in a te d , a t A by a v a r i a b le p o s i t i o n s h o r t
The p u l s e s r e f l e c t e d from th e s h o r t c i r c u i t i n th r e e
p o s i t i o n s , each s e p a r a te d by 2.3om s w ere re c o rd e d .
These r e c o r d in g s
w ere exam ined f o r l e a s t i n t e r f e r e n c e fiom s p u r io u s r e f l e c t i o n s and
c o n s ta n t tim e d i f f e r e n c e betw een p a i r s o f e q u a l a m p litu d e p o i n ts
o f p u ls e s r e f l e c t e d from p a i r s o f s h o r t c i r c u i t p o s itio n s *
I t was fo u n d t h a t w ith th e s h o r t c i r c u i t a d ja c e n t to th e
m agic to e th e r e c o r d in g s w ere q u i te u n s u ita b le f o r m easurem ents
b e c a u se o f an i n o r d i n a t e amount o f s p u rio u s r e f l e c t i o n s , w h ile
th o s e ta k e n a t 33*4 cms, f o r exam ple, do a p p e a r s u i t a b l e .
T hus, w h ile i t was n o t p o s s i b l e to red u c e th e s p u rio u s
r e f l e c t i o n s , th e m e a su rin g p o s i t i o n , A, was fo u n d such t h a t t h e y
h a d th e l e a s t a f f e c t on th e p u l s e s b e in g m easured.
The second s t e p in v o lv e d th e a c t u a l m easurem ents o f th e
r e f l e c t i o n i n t e r a c t i o n o f th e p u ls e w ith th e s te p p e d v/aveguide f o r
d if f e r e n t c u t- o f f fre q u e n c ie s .
The fo llo w in g i s a t y p i c a l re c o rd in g
p ro c e d u re .
The s h o r t c i r c u i t was p la c e d i n p o s i t i o n a t A and rem oved, th e n
r e p la c e d s e v e r a l tim e s and th e r e c o r d in g s o f th e r e f l e c t e d p u ls e s
su p e rim p o sed .
I n t h i s w ay, i t was hoped to gauge th e random e r r o r
c au sed by changing th e te r m in a t io n c o n ti n u a l l y d u rin g th e c o u rse o f
th e e x p e rim e n t.
The s te p p e d g u id e u n i t r e p la c e d th e s h o r t c i r c u i t
f o r r e c o r d in g and f i n a l l y th e s h o r t c i r c u i t p u ls e was r e c o rd e d a g a in .
T h is was r e p e a te d f o r each m o d if ic a tio n o f th e ste p p e d g u id e u n i t .
T hen, t h i s whole sequence was r e p e a te d f o r a d i f f e r e n t c a r r i e r
fre q u e n c y .
However, a s t h i s fre q u e n c y was in c r e a s e d , i t became i n c r e a s i n g l y
d i f f i c u l t to s e p a r a te s p u r io u s s i g n a l s from th e m ain p u ls e s i g n a l .
T h is i s b e c a u se th e group t r a v e l tim e f o r a g iv e n d i s t a n c e alo n g
th e p l a i n g u id e d e c r e a s e s w ith i n c r e a s in g fre q u e n c y due to th e
in c r e a s in g group v e l o c i t y .
B ecause o f t h i s and th e v a r i a t i o n o f th e
form o f th e p u ls e w ith c a r r i e r f re q u e n c y , i t was d e c id e d to stu d y
th e r e f l e c t i o n i n t e r a c t i o n o f a p u ls e o f f i x e d c a r r i e r fre q u e n c y
w ith th e ste p p e d g u id e f o r d i f f e r e n t b ro a d d im en sio n s and n o t v ic e
v e rsa .
I t was a n t i c i p a t e d from th e a n a l y s i s c o n ta in e d i n C h a p te r 3
t h a t r e f l e c t e d p u l s e s , w hich s u f f e r d i s t o r t i o n on i n t e r a c t i o n w ith
th e c u t - o f f g u id e , w ould a ls o s u f f e r an a p p a re n t change i n c a r r i e r
fre q u e n c y ( s e c t i o n 3 , 7 ) ) ,
T hus, th e g roup t r a v e l tim e f o r th e p u ls e
r e f l e c t e d from th e s h o r t c i r c u i t t e r m in a t io n to th e d e t e c t o r would be
d i f f e r e n t from t h a t o f th e p u ls e r e f l e c t e d from th e s te p p e d w aveguide
u n it*
A f u r t h e r e x p e rim e n t was d e s ig n e d to m easure t h i s e f f e c t .
The
r e c o r d in g s o f r e f l e c t i o n s from th e te r m in a t io n a t A w ere re c o rd e d i n
th e same m anner a s d e s c r ib e d p r e v io u s ly and w ere a ls o r e p e a te d a t a
p o i n t , B, 9cm s,, f u r t h e r away from th e magic t e e s e c t i o n .
The change
i n th e tim e o f t r a v e l f o r th e p u ls e r e f l e c t e d from th e s h o r t c i r c u i t
a t B w i l l be 2, A t ,
F o r th e p u ls e r e f l e c t e d from th e s te p p e d
waveguide u n it , t h i s w i l l be
At
+ At* where At* i s th e new group
t r a v e l tim e f o r 9om s., f o r th e d i s t o r t e d p u ls e w ith new a p p a re n t
!
c a r r i e r fr e q u e n c y , f ^ «
4., 7
D i.8 c u ^ 8 ^ n j q f _tW r e f l e c t ed p u ls e re c o rd in g s and m easurem ents
p e rfo rm e d on them
T h is e x p e rim e n t h a s b e en d e sig n e d " to m easure th e tim e o f
i n t e r a c t i o n o f a p u ls e on r e f l e c t i o n from a s e c t i o n o f g u id e , c u t - o f f
v /ith r e s p e c t to th e c a r r i e r fre q u e n c y ,"
T hus, some p o i n t on th e
r e c o rd e d p u ls e m ust be ch o sen a s a r e f e r e n c e p o in t f o r th e p u ls e i t s e l f
and th e tim e p o s i t i o n o f t h i s r e f e r e n c e p o in t w ith r e s p e c t to some
c o n v e n ie n t " t i n e z e ro " fo u n d .
The s a l i e n t p o in t o f each d e te c te d
p u ls e was i t s p e ak and i n th e r e g io n o f t h i s p e a k , th e r e c o r d in g
v/as sy m m e tric a l i n a lm o st a l l c a s e s .
I t i s p ro p o se d t h a t th e tim e
p o s i t i o n o f th e p eak w i l l be ta k e n a s th e r e f e r e n c e p o i n t o f th e
r e c o rd e d p u l s e .
T here i s some e x p e rim e n ta l jU ( S tif ic a tio n f o r t h i s
i n th e r e c o r d in g s o f p u l s e s r e f l e c t e d from th e th r e e c l o s e l y sp aced
sh o rt c ir c u it p o s itio n s .
T hese r e c o r d in g s shov; e v e n ly sp a ce d p e a k s ,
s e p a r a te d i n tim e by th e group t r a v e l tim e s , p r e d i c t e d from a
knowledge o f th e c a r r i e r fre q u e n c y .
—120*"*
(4-c 7)
I t fo llo w s fro ri t h i s t h a t a c o n v e n ie n t " t i n e z e ro " i s th e
p o s i t i o n o f th e p e a k o f t h e p u ls e r e f l e c t e d from th e s h o r t c i r c u i t
te r m in a tio n ^
The d i f f e r e n c e s betw een th e av erag e re c o r d e d p o s i t i o n s
o f th e p e a k s o f p u l s e s r e f l e c t e d from th e s h o r t c i r c u i t and s te p p e d
w aveguide u n i t s r e s p e c t i v e l y was o b ta in e d .
T hese d e la y tim e s f o r
th e v a r io u s s te p p e d g u id e u n its ^ m easured a t 25cms. and 34cms, from
th e m agic t e e , f o r f^ = 5o95 GHz a re g iv e n i n th e s i x t h and se v e n th
rows o f th e t a b l e o f t y p i c a l m easurem ents p r e s e n te d i n f i g u r e 26(116)
F ig u r e s 27a(l21 ) , b ( l 2 2 ) , c(l2 3) c o n ta in g ra p h s o f th e s e m easured
d e la y tim e s f o r v a r io u s ste p p e d g u id e u n i t s f o r t h r e e d i f f e r e n t
c a r r i e r fre q u e n c ie s .
The a b c i s s a i s th e r a t i o o f th e c u t - o f f
f r e q u e n c ie s o f th e m o d ifie d and u n m o d ifie d g u id e s r e s p e c t i v e l y .
T h is sh o u ld f a c i l i t a t e
com parison w ith t h e o r e t i c a l l y p r e d i c t e d tim e s .
A lso g iv e n on th e s e f i g u r e s a r e th e m easured tim e d e la y s f o r th e
u n m o d ifie d p u ls e o v e r a n e x t r a I8cm s, p a th le n g th and th e p r e d i c t e d
tim e o f t r a v e l and a ls o th e d is ta n c e o f A from th e m agic t e e .
The
f u l l cu rv e on f i g u r e s 27a , b and o and th e q u a n t i t i e s c o n ta in e d i n
th e f i f t h ro v ^ f f i g u r e 26 g iv e th e p r e d i c t e d r e f l e c t i o n i n t e r a c t i o n
tim e c o rre sp o n d in g to th e fre q u e n c y , f ^ , and th e r a t i o ^ ^ 2 / ^ 01"
I t m ust be s t r e s s e d t h a t a p o i n t on t h i s curve does n o t r e p r e s e n t
d i r e c t l y th e r e f l e c t i o n i n t e r a c t i o n tim e o f a p u ls e b u t th e tim e
c o rre s p o n d in g to th e p a r t i c u l a r fre q u e n c y , f ^ .
The p e n u ltim a te and
u l tim a te rows o f f i g u r e 26 l i s t th e d e la y tim e f o r th e e x t r a 18cms.
p a th l e n g th f o r th e i n c i d e n t p u ls e and th e p u ls e i n t e r a c t i n g w ith
th e m o d ifie d s e c t i o n .
F ig u r e s 28(12<i) and 29(125) show t y p i c a l
s e t s o f r e c o r d in g s f o r a low and h ig h c a r r i e r fre q u e n c y p u l s e .
"1 2 1 .
(4.7)
n seos
1.4
M e a su r e m e n ts ta k e n
4* 2 5 o m s
at
X 34'0ms
O 54cm s
P r e d ic te d
cu rve
G rou p t r a v e l t im e f o r 1 8 o m s;
M easu red :
* 86 hh . 0 1
n seos.
P r e d ic te d :
.8 7
n seos
P ifiu r e
27a.
The g ra p h o f th e
r e fle c tio n
tim e ,
th e
tj^ . v e r s u s
in te r a c tio n
r a tio , o f
th e
c u t-o ff
f r e q u e n c ie s o f t h e m o d ifie d an d u n m o d ifie d g u id e s .
= 5'95G H z.
.122 -
(4.7)
.6
.5
.4
tj^ i n
n seos
•
2
•1
1.Ü
I M e a su r e m e n ts
— P r e d ic te d
ta k e n a t
curve
2 5om s,
(V = ^ o 1 >
=
G roup t r a v e l t im e f o r 1 8 cm s;
M easu red ;
* 8 5 + .0 1 n s e o s .
P r e d ic te d ; #86
n seos.
P i^ u r e
27b,
The arap h o f th e
^ tj^ , v e r s u s t h e
r e fle c tio n
r a tio
o f th e
m o d ifie d
f o r f^
rs 6 , 0 5 G H z ,
o f th e
in te r a c tio n
tim e ,
c u t - o f f fr e q u e n c ie s
an d u n m o d ifie d g u id e s
( f ^ V f )
-1-23-
(4.7)
t^ in
n seos
m e a s u r o m e n ts ta lc e n a t
• P r e d ic te d ou rve
25om a
G ro u p t r a v e l t i m e f o r 1 8cm s*
m easu red :
. 8 ^ .0 1 n s e o s
p r e d ic te d : .8 2
n seos
( f o / f ^ ^ ) = "1*48
P ifc u r e
27o,
The ^ ra p h o f
th e
r e fle c tio n
in te r a c tio n
tim e, tj^. versu s the r a tio o f the c u t-o ff
freq u en oies o f the m odified and unmodified
Guides
f^
= 6 .3 5 G H z .
(4 . 7 )
—12/f—
a)
a m p litu d e
L—►t i m e
1 cm 5
*81
n secs
ty j/j
/
mF ig u r e
28.
XÏ* r e c o r d i n g s , t a k e n f r o m t h e s a m p l i n g o s c i l l o s c o p e .
of* t h e r e c t i f i e d e n v e l o p e s o f r e f l e c t e d p u l s e s f r o m
a)
b)
s h o r t c i r c u i t p o s it io n s s e p a r a te d b y 2*5om s
s h o r t c i r c u i t (OC ) a n d m o d i f i e d g u i d e s o f c u t - o ! ( f
f r e q u e n o i e s 8 . 1 l ( p ) , ? .3 4 ('» ' ) , 6 . 7 ( 6 ) , 6 . 4 * ( E )
a n d 6 . 1 2 ( 5 )& H z .
(4 . 7)
-12 5 -
a) sh o rt c ir c u it
p o s itio n s se p a r a te d
b y 2 .5 c m s
a m p litu d e
ï —) t i m e
1 cm a
•81
f
o
n seos
=
6.35GHz
b ) s h o r t c i r c u i t (c C .)
an d m o d ifie d g u id e s o f
c u t - o f f fr e q u e n c ie s
8.11(P),
7.34(7),
6.44(5),
6.7 ( e )
1
7.69(4 ) ,
6.99(f ) ,
6.12(tv) ,
I
F ig u r e
29 • R e c o r d in g s
o f th e
r e c tifie d
e n v e lo p e s o f
r e fle c te d
P u l si
26*i f .8
(ii-« 8)
C o n tro l e x p e riia e n t f o r r e f l e c t i o n from th e lo n g s te p p e d g u id e u n i t ,
The p re s e n c e o f th e d i s c o n t i n u i t y i n th e form o f a sudden s te p
i n th e w aveguide v d .ll i t s e l f g iv e r i s e to r e f l e c t i o n s .
I t has been
assum ed t h a t th e r e f l e c t i o n and h en ce th e r e f l e c t i o n i n t e r a c t i o n tim e
i s th e r e s u l t o f th e change i n p r o p a g a tio n c o n s ta n t o f th e m o d ifie d
g u id e s e c t i o n .
T h is e x p e rim e n t was d e s ig n e d to c l a r i t y t h i s p o i n t .
The aim o f i t i s to r e a l i s e th e sudden s te p i n th e w aveguide w ith o u t
th e change i n p r o p a g a tio n c o n s ta n t a c r o s s th e s t e p .
T h is was a c h ie v e d
by p l a c in g m e ta l shim s a t th e p o s i t i o n A , fo llo w e d by a p l a i n g u id e ,
2}-3cms. lo n g and a m atch ed lo a d .
sh im s.
R e c ta n g u la r h o le s w ere o u t i n th e s e
The h e ig h t was f i x e d and c o rre sp o n d e d to t h a t o f WG-14 wave­
g u id e w h ile th e w id th s c o rre sp o n d e d a p p ro x im a te ly to th e re d u c e d b ro a d
d im en sio n s o f th e s te p p e d w aveguide u n i t s ,
F ig u re 3 0a(127) shows th e
shim i n p o s i t i o n on th e p l a i h w aveguide w h ile f i g u r e 30b c o n ta in s a
t a b l e o f th e re d u c e d b ro a d d im e n sio n s and c o rre s p o n d in g c u t - o f f
fre q u e n c ie s .
The b ro k e n l i n e i n f i g u r e 30a i n d i c a t e s th e c r o s s -
s e c t i o n o f th e p l a i n g u id e .
F ig u re
(128) shows r e c o r d in g s o f th e
r e c t i f i e d e n v e lo p e s o f p u l s e s r e f l e c t e d from th e s h o r t c i r c u i t and th e
sh im s, f o r f^ =
T hese may be compared w ith f i g u r e 28(i24 )
and 29( 125) f o r th e r e c o r d in g s o f p u l s e s r e f l e c t e d from th e ste p p e d
w aveguide u n i t s .
I t i s a p p a re n t t h a t a m ea su rab le tim e d e la y i s
n o t p r e s e n t i n th e shim r e c o r d in g s .
T h e re fo re , i t w i l l be c o n clu d ed
t h a t th e ’*r e f l e c t i o n i n t e r a c t i o n d e la y tim e s ’* deduced may be a t t r i b u t e d
to th e r e f l e c t i o n i n t e r a c t i o n o f th e p u ls e w ith th e w hole o f th e
m o d ifie d g u id e and n o t w ith th e s te p a lo n e ,
4*9
E v a lu a tio n o f th e m ea su rin g system and r e s u l t s o b ta in e d .
One o f th e a d v a n ta g e s o f p e rfo rm in g th e e x p e rim e n ts i n W&I4
(ii-.9)
- 1 27'
The b ro k e n l i n e i n d i c a t e s th e c r o s s - s e c t i o n o f th e u n m o d ifie d
g u id e , ABCD. The f u l l l i n e i n d i c a t e s th e c r o s s - s e c t i o n ,
PQES, o f th e h o le i n th e shim .
ja.
D iagram show ing th e shim i n f r o n t o f th e f la n g e
o f th e p l a i
1
1
Reduced b ro a d
d im en sio n i n
cms.
2 .9 5
c u t-o ff
fre q u e n c y o f
H , mode i n
GHz.
5 .0 8
P u ls e
re c o r d in g on
F ig u re 31 «
P ig u re 30b
I
I
1
I
2.3 5 5
2 .2 6
2.16
2.055
1 .9 6
1 .8 5 5
6 .1 2
6 .3 7
6 .6 lf
6 .9 4
7 .2 3
7 .6 5
8 .0 8
6
T)
E
e
6
Y
2 .4 5
i
/<
I
T ab le o f re d u c e d b ro a d d im o n ^ o n s o f th e shim s an d
t h e i r c o rre s p o n d in g c u t - o f f
P
- 128-
(4 . 9)
a m p litu d e
.tim e
F ig u r e
31#
1 cm s
,8 1
n seos
R e c o r d in g s
o f p u ls e s
(see
o f
th e
r e c tifie d
r e fle c te d
fig u r e
30b)
fr o m
e n v e lo p e s
th e
and s h o r t
s h im s
c ir c u it
(cc).
-1 2 9 -
( 4 .9 )
w aveguide was t h a t i t a llo w e d th e d e t e c t i o n o f a l a r g e amount o f
s i g n a l , f o r in p u t to th e sam p lin g o s c i l l o s c o p e .
? o r exam ple, th e
p eak d e f l e c t i o n on th e o s c illo s c o p e c au sed by th e r e c t i f i e d en v elo p e
o f a p u ls e from th e t . w . t . was a p p ro x im a te ly f i v e tim e s g r e a t e r th a n
th e d e f l e c t i o n due to th e r e c t i f i e d c.w . in p u t to th e t . w . t .
A ls o ,
th e m agic t e e system a llo w e d d e t e c t i o n o f th e r e f l e c t e d p u ls e a lo n e
i n s t e a d o f th e i n c i d e n t and r e f l e c t e d p u ls e s t o g e t h e r , as i n a p ro b in g
sy stem , w ith t h e i r m utual i n t e r f e r e n c e .
However, th e m ain d is a d v a n ta g e o f th e system was t h a t th e
r e f l e c t i o n i n t e r a c t i o n tim e c o u ld n o t be m easured d i r e c t l y a d ja c e n t
to th e s te p p e d g u id e u n i t ( s e e s e c t i o n ( 4 v 6 ) ) .
T h is meant t h a t any
change i n th e g roup t r a v e l tim e o f th e d i s t o r t e d p u ls e from A b a c k to
th e d e t e c t o r com pared w ith th e g ro u p t r a v e l tim e c o rre sp o n d in g to th e
u n d i s t o r te d p u ls e r e f l e c t e d from th e s h o r t c i r c u i t , Tfas u n a v o id a b ly
added a l g e b r a i c a l l y to th e tim e w hich w ould have b e en m easured
d i r e c t l y a d ja c e n t to th e c u t - o f f s e c tio n .
F o r t h i s r e a s o n , th e tim e
d e la y deduced from th e r e c o r d in g s w i l l be c a l l e d th e " r e f l e c t i o n
i n t e r a c t i o n d e la y tim e " to d i s t i n g u i s h i t from th e " r e f l e c t i o n
i n t e r a c t i o n tim e " p r o p e r .
The m ethod o f ded u cin g t h i s d e la y tim e
h a s b e e n d e s c r ib e d i n th e p r e v io u s s e c t i o n .
T h e r e fo r e , b e a r in g i n mind th e l i m i t a t i o n s o f th e m ea su rin g
system and th e c o n tr o l e x p e rim e n t w ith th e sh im s, i t can b e s a i d t h a t
i n t e r a c t i o n d e la y tim e s f o r p u l s e s r e f l e c t e d from lo n g s e c t i o n s o f
gu id e c u t - o f f w ith r e s p e c t to th e c a r r i e r fre q u e n c y have b e e n m easured
and com pared.
F o r e ach c a r r i e r fre q u e n c y , a q u ite d e f i n i t e t r e n d
h a s b een e s t a b l i s h e d v i z . , a s th e r a t i o o f th e c u t - o f f f r e q u e n c ie s
o f th e m o d ifie d and u n m o d ifie d g u id e s i n c r e a s e s , th e m easured
-1 3 0 -
(i;..9)
r e f i e o t io n i n t e r a c t i o n d e la y tim e d e c r e a s e s .
i n a n o th e r way.
T h is may h e s t a t e d
As th e r a t i o o f th e c a r r i e r fre q u e n c y to th e c u t­
o f f fre q u e n c y o f th e m o d ifie d g u id e i n c r e a s e s to w a rd s u n i t y , th e
m easured d e la y tim e a ls o i n c r e a s e s ,
E x ar^ in a tio n o f f i g u r e 26(116 )
r e v e a l s t h a t f o r th e s m a lle r v a lu e s o f f f ^ ^
f^
( o r v a lu e s o f
u n i ty - see th e t h i r d row) t h e r e i s a change i n th e
group t r a v e l tim e o f th e r e f l e c t e d p u ls e com pared w ith th e i n c i d e n t
p u ls e ( p e n u ltim a te and u l ti m a t e ro w s).
These v a lu e s c o rre sp o n d to
th e more d i s t o r t e d p u ls e s o f th e r e c o r d in g s shown i n f i g u r e ^8(124 ) .
F o r th e l e s s d i s t o r t e d p u l s e s corresiD onding to l a r g e r v a lu e s o f
( o r s m a lle r v a lu e s o f f ^
deduced.
such a change c a n n o t he
An e x p la n a tio n o f th e s e tr e n d s may he fo u n d on exam ining
th e t h e o r e t i c a l l y p r e d i c t e d c u rv e s o f f i g u r e s 32(131 ) , 33(132) and
3 4 (1 3 3 ).
F ig u re 32(131 ) c o n ta in s a g rap h o f r e f l e c t i o n i n t e r a c t i o n
tim e v e rs u s f re q u e n c y .
Each c u rv e c o rre sp o n d s to a p a r t i c u l a r
m o d ifie d g u id e and i s l a b e l l e d hy i t s c u t - o f f fre q u e n c y and a ls o
th e r a t i o o f i t s c u t - o f f fre q u e n c y to t h a t o f th e u n m o d ified g u id e
E x p re s s io n (3 * 6 .6 ) (d ed u ced from (3 * 5 .6 )) h a s h e e n u se d to p r e d i c t th e
r e f l e c t i o n i n t e r a c t i o n tim e s f o r f r e q u e n c ie s helow c u t - o f f h u t th o s e
f o r f r e q u e n c ie s ahove c u t - o f f a re n o t in c lu d e d .
I t was fo u n d i n th e
l a t t e r i n s t a n c e , t h a t th e v a lu e s o f tim e o s c i l l a t e d g r e a t l y so th e y
a r e o m itte d from f i g u r e 32(l 31 ) f o r th e sake o f c l a r i t y .
How ever,
f i g u r e 33(132) shows th e v a r i a t i o n o f r e f l e c t i o n i n t e r a c t i o n tim e
ahove and h elow c u t - o f f f o r th e m o d ifie d g u id e , f^ g = 6,4^.G'Hz.
F ig u re 34(133) shows th e v a r i a t i o n o f th e m odulus o f th e r e f l e c t i o n
c o e f f i c i e n t , | B/A| , f o r t h i s s te p p e d g u id e u n i t , ahove and helow
i t s c u t - o f f fre q u e n c y ,
I t w i l l he n o te d t h a t th e m odulus o f th e
- 1 31-
ill
1
IT
S 0 C5 #
It
f ginCHz
Î
I
fre q u e n c y
F ig u r e 3 2 .
i n GHz
The g ra p h o f th e t h e o r e t i c a l l y p r e d i c te d r e f l e c t i o n
i n t e r a c t i o n tim e ,
, v e r s u s fre q u e n c y , f o r g u i d e s .
3Qcms lo n g , b u t o f d i f f e r e n t c u t - o f f f r e q u e n c i e s , f
8
o2
"W
- 132 -
(4.9 )
f
S M S ilë
-»+f-4- k
-4—
»I •. »-* «-*-4*
a
isM f
ec8
;i±:
:
m m #
2.10
i n G-Hz.
F ig u r e 35» The g ra p h o f th e t h e o r e t i c a l l y p r e d i c t e d r e f l e c t i o n
i n t e r a c t i o n tim e , t p . v . f r e q u e n c y . sho;Yiy!; th e
o s c i l l a t o r y r e g i o n &>ove c u t - o f f ,
= 6.440H z)
-1 0
-133-
(4 .9 )
1.0
.9
.8
.7
.6
iB/Al
ta+ttri
rMttTrrf
.3
.2
fe] I ®
6 .4
F ig u re 34.
MilfrH
fre q u e n c y
6 .8
i n GHz
The g ra p h o f th e t h e o r e t i c a l l y p r e d i c te d
m odulus o f th e r e f l e c t i o n co e f f i c i e n t , |B / a|.
f o r th e m o d ifie d g u id e , f \ = 6 . W^GHz. and~
30cms lo n g .
^
-1 3 4 -
(4 .9 )
r e f l e c t i o n c o e f f i c i e n t f o r th e s e v e r y lo n g g u id e s i s u n i t y i n th e
c u t - o f f range
f^^
<
f
from f i g u r e 32(131)
< f^ g .
f o r a g iv e n c a r r i e r fre q u e n c y i n th e
c u t - o f f re g io n o f th e m o d ifie d g u id es^ f o r example ^ f^ = 5«95GHz
i t can ho soon t h a t a s th e r a t i o , ^ c 2 '^ ^ o 1 ' ( th e q u a n t i t y i n
b r a c k e ts on each cu rv e ) d e c r e a s e s , so th e r e f l e c t i o n i n t e r a c t i o n
tim e i n c r e a s e s .
T h is i s i n a c c o rd a n c e w ith th e m easured t r e n d
d e s c r ib e d above i n t h i s s e c t i o n .
A ls o , an in c r e a s e i n group t r a v e l tim e s f o r th e lo w e r v a lu e s
o f f g y /f^^ was fo u n d and a d is c r e p a n c y betw een th e m easured and
p r e d i c t e d i n t e r a c t i o n tim e s ,
A p r e c i s e e x p la n a tio n o f t h i s t r e n d
w ould n eed com plete in f o r m a tio n on th e ph ase and a m p litu d e o f th e
i n i t i a l p u ls e .
H ow ever, i t i s p o s s i b l e to p u t fo rw a rd a q u a l i t a t i v e
e x p la n a tio n w ith c e r t a i n a s s u m p tio n s .
S ince th e v a lu e o f th e
c a r r i e r fre q u e n c y o f t h e p u ls e may be u se d to p r e d i c t th e group
t r a v e l tim e o f th e i n i t i a l p u ls e o v e r a g iv e n d i s t a n c e i n th e
u n m o d ified g u id e w ith some a c c u ra c y , i t may be assum ed t h a t th e
a m p litu d e d i s t r i b u t i o n i n th e f re q u e n c y domain h a s a maximum v a lu e
a t t h i s fre q u e n c y .
Hov/ever, n o th in g can be s a i d c o n c e rn in g th e
fre q u e n c y d i s t r i b u t i o n a b o u t t h i s c a r r i e r fre q u e n c y .
However, a s
I
th e c a r r i e r fre q u e n c y v;as f i x e d i n a p a r t i c u l a r e x p erim e n t and th e
m o d if ic a tio n s to t h e g u id e v a r i e d , th e fre q u e n c y d i s t r i b u t i o n d u rin g
t h i s tim e can a ls o be assum ed c o n s t a n t .
W ith t h i s h y p o t h e t i c a l
c o n s ta n t d i s t r i b u t i o n ab o u t th e f i x e d c a r r i e r f re q u e n c y , i t i s
a p p a re n t t h a t a s th e r a t i o f ^ ^ /
decreases ( o r f ^ / f i u - o r e a s e s
to w a rd u n ity ) more and more o f i t w i l l be above th e c u t - o f f fre q u e n c y
i n th e o s c i l l a t i n g r e g io n shov/n i n b o th f i g u r e s 3 3 (l3 2 ) nnd 34( 133 ) #
-1 3 5 -
(4 .9 )
The z e ro s o f th e R e f le c t i o n c o e f f i c i e n t \S>/à.\
ahove c u t - o f f i n
f i g u r e 34 o c o u r a t f r e q u e n c ie s w hich c o rre sp o n d to th e l e n g t h o f
m o d ifie d g u id e , a , b e in g an i n t e g r a l nuiuber o f h a l f w a v e le n g th s .
Thus I B/il I , th e modulus o f th e t r a n s f e r f u n c t i o n o f t h i s system and
m u ltip ly in g f a c t o r f o r th e fre q u e n c y d i s t r i b u t i o n f u n c t i o n i s no
lo n g e r u n i t y and i t i s to be e x p e c te d t h a t th e s e p u ls e s w i l l be
d is tu rb e d .
T h is i s shown q u i te d i s t i n c t l y i n th e p u ls e r e c o r d in g s
o f f i g u r e r 2 8 b (l2 4 ) and 29^(1250 «
F o r th e s e d i s t o r t e d p u l s e s , a
d i r e c t com parison betw een th e m easured d e la y tim e and th e r e f l e c t i o n
i n t e r a c t i o n tim e p r e d i c te d from a Imowledge o f th e i n c i d e n t c a r r i e r
fre q u e n c y c a n n o t be made#
The i n c r e a s e i n group t r a v e l tim e can be
th o u g h t o f as an a p p a re n t d e c re a s e i n c a r r i e r fre q u e n c y .
The m easured i n t e r a c t i o n tim e i s th e a lg e b r a i c sum o f th e
r e f l e c t i o n i n t e r a c t i o n tim e and th e d i f f e r e n c e i n tim e o f t r a v e l
o f th e m o d ifie d and u n m o d ified p u ls e s i n th e p l a i n g u id e to th e
d e te c to r.
T hus, f o r th e d i s t o r t e d p u l s e , th e lo n g e r th e p a th w hich
i t t r a v e r s e s th e l a r g e r w i l l be th e m easured d e la y tim e .
T hat t h i s
i s so e x p e r im e n ta lly can be fo u n d on e x a m in a tio n o f th e s i x t h and
s e v e n th rows o f f i g u r e s 26(f16) f o r
m easurem ents ta k e n a t 23 and
34oms.
I n f i g u r e 27a(121 ) i t w i l l be n o te d t h a t th e d e la y tim e s
deduced f o r p u l s e s o f c a r r i e r fre q u e n c y 3*96GHz a t 34^ oms a re n o t
l a r g e r th a n t h o s e o f 23oms and 34gi-1s, a s m ight be e x p e c te d .
such a co m p ariso n i s n o t v a l i d .
However,
The m easurem ents a t 23 and 34oms
w ere made w ith e x a c tl y th e saiae p u l s e .
Those a t 34cms. w ere made
w ith th e same c a r r i e r fre q u e n c y c e r t a i n l y b u t n o t d u rin g th e same
ex p erim e n t so t h a t o t h e r c h a r a c t e r i s t i c s o f th e p u ls e were n o t th e
-136-
saine a
(4.9)
I t was f o r t h i s same s o r t o f re a s o n t h a t th e s e r e f l e c t i o n
e x p e rim e n ts w ere made w ith a f i x e d c a r r i e r fre q u e n c y p u ls e and
s e v e r a l m o d ifie d s e c t i o n s and n o t v ic e v e r s a .
t h a t f o r th e l a r g e r v a lu e s o f f ^ ^ / f ^ ^
I t v d l l be e x p e c te d
( o r s m a ll e r f ^ / ^ ^ ^ ^ r e
th e m ajo r p a r t o f th e fre q u e n c y d i s t r i b u t i o n v d l l l i e i n th e c u t - o f f
ra n g e j i n w hich 1^^/â!
i s a p p ro x im a te ly u n i t y , t h e r e w i l l be n e g l i g i b l e
d i s t o r t i o n o f th e r e f l e c t e d p u ls e and l i t t l e
d i f f e r e n c e betw een th e
e x p e r im e n ta lly m e a su re d and th e p r e d i c t e d tim e .
10
F u r th e r r e f l e c t i o n i n t e r a c t i o n experim e n ts w ith lo n g and s h o r t
mo d i f i e d g u id e s
I n th e r e f l e c t i o n e x p e rim e n ts , d e s c rib e d i n th e p r e v io u s s e c t i o n s ,
th e le n g th o f m o d ifie d s e c t i o n was 30cms, w hich r e p r e s e n te d a v a r i a t i o n
from 6 ,1 2 to 8.11 c u t o f f w a v e le n g th s f o r th e m o d if ic a tio n s in tr o d u c e d .
B ecause o f th e a t t e n u a t i o n o f s i g n a l on tr a n s m is s io n th ro u g h a
m o d ifie d s e c t i o n c u t - o f f w ith r e s p e c t to th e c a r r i e r fre q u e n c y o f th e
s i g n a l , s u i t a b l e l e n g t h s o f m o d ifie d g u id e u se d f o r tr a n s m is s io n
e x p e rim e n ts were n o t e x p e c te d to ex ce ed a c u t - o f f w a v e le n g th .
To
b r id g e th e gap b e tw ee n th e s e t r a n s m i s s io n e x p e rim e n ts w ith s h o r t
m o d ifie d s e c tio n s and th e r e f l e c t i o n e x p e rim e n ts w ith lo n g o n e s , i t
was d e c id e d to i n c lu d e r e f l e c t i o n e x p e rim e n ts f o r m o d ifie d s e c tio n s
o f f i x e d c r o s s - s e c t i o n b u t v a r i a b l e le n g th , f o r a g iv e n c a r r i e r
fre q u e n c y .
The le n g t h s chosen w ere 30ciûs, 3»2cms and 1 ,6 o r 1,3Gms
Sind a c a r r i e r fre q u e n c y o f
th e tr a n s m is s io n e x p e r im e n ts ) ,
(The 3«2cms l e n g th was u s e d f o r
The re c o rd e d p u ls e s f o r m o d ifie d
w id th s o f 2.45cm s ( .6 3 ^Cg) and 2,,05cms ( ,7 8
f i g u r e s .33(137-) ^hd •36(138 )■ r p s p o .c tiv e ly .j
a re shown i n
( 4 . 10)
-1 3 7 -
I
1
I
I
*
I
‘
(
Î
5ÎS:SSSS^îS;.^Bs(iM *UΣvSË3Sôa£M fs!»^î^ "
88 n seo s
1 cm
f(,
F ig w e 35
= 5 .9 5 G H z
R e o o r c lin g s o f
r e fle c te d
th e
fr o m
r e c tifie d
sh ort
c ir c u it
e n v e lo p e s o f p u ls e s
(<X. )
g id .d e 3 o f l e n g t h s
30cm s ( p ,
3 .2 o m s
f
("Y ,
1.5oras ( 6
,
^2
«
6 .1 2 & H z ) .
f
2
= 6 ,0 5 G H z )
f
2
= 6 .0 1 G H z ) .
a n d r e d u c e d b r o a d d im e n s io n o f
a p p r o x im a t e l^ y ' 2#!*.5om s#
and m o d ifie d
'
^
(4 . 10)
138-
O
#'
#*W
.M
iLU*M
#W
î]^^3^Sl^W
%
e:%
%
pa;5%
i{W
kÆ
2M
S^nS%
gup.^VLirig34pïr$-GCW
tp%
r;($V«igk%
'3p,:2n;$ce;,9K^
! ’ î *'
8*
J j
a m p litu d e
V tim e
88 n seô s
= 3.95G -H Z
F ifiu r e
36,
R e c o r d in g s o f
th e
p u ls e s r e fle c t e d
r e c tifie d
fr o m
sh ort
e n v e lo p e s
c ir c u it
and m o d ifie d g u id e s o f le n g t h s ♦
30 oms. ( p .
^c2
3.2cms. ( Y ,
^o2
1 . 6 oms ( 6 ,
^o2
and red u ced b road
a p p r o x im a te ly
d im e n s io n o f
2 . 05 c m a .
o f
( OC)
2l ,11
Q u a l i t a t i v e E v a lu a tio n o f th e r e s u l t s o f th e s e f u r t h e r
r e f l e c t i o n e x p e rim e n ts .
G-raphs o f th e t h e o r e t i c a l l y p r e d i c te d m odulus o f th e r e f l e c t i o n
c o e f f i c i e n t and r e f l e c t i o n i n t e r a c t i o n tim e s v e r s u s fre q u e n c y f o r
m o d ifie d g u id e s o f g iv e n c u t - o f f fre q u e n c y and le n g t h s 30, 3 .2 and
1*6 o r 1*5cms*^ a r e p r e s e n te d i n f i g u r e 37 to 4 0 .
Those c u rv e s i n
f i g u r e s 3 7 (M 0) and 38(14-1 ) c o rre sp o n d to a c u t - o f f fre q u e n c y f^g"%
7G-Hz and th o s e i n f i g u r e s 39(142_) and 40(143) to f 6 . 0 5 G - H Z *
The
c a r r i e r fre q u e n c y i s a ls o i n d i c a t e d on th e g ra p h s .
F o r th e s e c t i o n s w ith f _ ^ 70Hz, i t v / il l be n o te d t h a t th e
c a r r i e r fre q u e n c y i s n o t n e a r c u t - o f f .
a r e i n d ic a t e d on th e p a r t i c u l a r c u r v e ) .
(The e x a c t c u t - o f f f r e q u e n c ie s
W ith r e g a r d to f i g u r e 3 7 ^
i t w i l l be e x p e c te d t h a t th e d i s t o r t i o n o f th e r e f l e c t e d p u ls e s w i l l
n o t be g r e a t , becom ing g r e a t e r , how ever, w ith d e c r e a s in g le n g t h .
W ith t h i s i n m ind and w ith f u r t h e r r e g a r d to f i g u r e 38, i t i s to be
e x p e c te d t h a t t h e r e f l e c t i o n i n t e r a c t i o n tim e s w i l l be a p p ro x im a te ly
th e same.
The r e c o r d in g s o f f i g u r e 3^ b e a r t h i s o u t . F o r th e s e c tio n s
o f w id th
2.43cm s o r f ^ ^ 6G-Hz, th e c a r r i e r fre q u e n c y i s v e iy n e a r
c u t-o ff.
The g rap h c o n ta in e d i n f i g u r e 39 shows a r a p id change o f
modulus o f th e r e f l e c t i o n c o e f f i c i e n t f o r th e s h o r t e r s e c tio n s i n th e
r e g io n o f th e c a r r i e r fre q u e n c y .
The r a p i d l y o s c i l l a t i n g r e g io n above
c u t - o f f ( s e e a s i m i l a r curve i n f i g u r e 34) f o r th e 30cms lo n g s e c t i o n
h a s been o m itte d .
T hus, much d i s t o r b i o n i s to be e% pected a s th e
r e c o r d in g s i n f i g u r e 35 show.
Thus th e p r e d i c t e d i n t e r a c t i o n tim e s
c o rre s p o n d in g t o th e c a r r i e r fre q u e n c y o f th e i n c i d e n t p u ls e and th e
m easured d e la y tim e s c an n o t be com pared.
However, i t w i l l be
m en tio n ed i n p a s s in g t h a t th e i n c r e a s e i n i n t e r a c t i o n tim e from th e
s h o r t e s t to th e l o n g e s t s e c t i o n shown i n f i g u r e 40 i s fo u n d
—14-0 —
(4.11)
1 .0
riîTiîTH
.9
1
.8
i
|B/A|iiin:'
üî!
Hi
curve f 2 in&Hz le n g t h
6.99
P 7.14
T 7.14
.1
5
F ig u r e 3 7 .
lili
iiiü
fre q u e n c y
cms.
i n GHz,
The grap h o f th e t h e o r e t i c a l l y p r e d i c te d
m odulus o f th e r e f l e c t i o n c o e f f i c i e n t , |B /A |, v e rs u s
fre q u e n c y f o r 3 m o d ifie d s e c t i o n s o f a p p ro x im a te ly th e
same c r o s s - s e c t i o n b u t d i f f e r e n t l e n g t h s .
—I Zf1—
( 4 . 11)
.4
H§8is!s:!i
i
!s»S»!S8»;8s
I
1
I
5
a
-
f o2
ft
T
Fiprure 38,
freq u en cy
6
6 . 99G-H2 ,
3O0111S lo n g .
7.14GHz,
7.14GHz,
3 . 2 cms
1 , 6 cms
i n &Hz
The g rap h o f th e t h e o r e t i c a l l y p r e d ic te d r e f l e c t i o n
i n t e r a c t i o n tim e ,
f o r 3 m o d ifie d s e c tio n s o f
a p p ro x im a te ly th e same c r o s s - s e c t i o n b u t d i f f e r e n t
le n g th s .
(4 .1 1 )
—14-2—
curve f ^ o inG-Hz«len p ;th
“ cms
1.0
It
tl!i m;
i
5
F ig u r e 39»
fre q u e n c y
i n G-Hz
The # ra p h o f th e t h e o r e t i c a l l y p r e d i c t e d m odulus o f th e
r e f l e c t i o n c o e f f i c i e n t f o r 3 m o d ifie d s e c t i o n s o f
a p p ro x im a te ly th e same c r o s s - s e c t i o n h u t d i f f e r e n t
le n g th s .
(4.11)
-1 4 3 ^
cu rv e f ^ in G-Hz le n g t h
-------o2----------
•4
.3
•t
, r
' in
n secs
•
2
SSSSSii iSSSSSSSSSISSSSS
.1
3
F ig u r e 40
fr e q u e n c y
o
hMTi n GHz
7
The g ra p h o f th e t h e o r e t i c a l l y p r e d i c te d r e f l e c t i o n
i n t e r a c t i o n tim e ,
v e r s u s fre q u e n c y f o r 3 m o d ifie d
s e c t i o n s o f a p p ro x im a te ly th e same c r o s s - s e c t i o n b u t
d iffe re n t le n g th s .
-1 4 4 -
(4 .1 1 )
e x p e r im e n ta lly i n th e r e c o r d in g s o f f i g u r e 33-
Hov/ever^ th e m easured
tim e s a re a l l g r e a t e r th a n p r e d i c t e d f o r th e i n c i d e n t c a r r i e r fre q u e n c y .
T h is c o u ld be a t t r i b u t e d to a p o s s ib l e s h i f t to a lo w e r c a r r i e r
fre q u e n c y , r e s u l t i n g i n a l o n g e r group t r a v e l tim e to th e d e t e c t o r .
However, e x p e rim e n ts to v e r i f y t h i s p r o p o s i t io n w ere n o t c a r r i e d o u t
as in s e c tio n ( 4 .7 ) .
4 .1 2
Transm i s s io n e x p e rim e n ts w ith sho r t s te p p e d wave g u ide s e c t i o n s
The d e t e c t i n g system i n th e s e e x p e rim e n ts y i e l d e d v e ry l i m i t e d
in f o r m a tio n and a com parison b etw een t h e o r e t i c a l l y p r e d i c t e d and
m easured r e s u l t s can be made c o n f id e n tly o n ly when t h e r e i s l i t t l e
d i s t o r t i o n o f th e r e f l e c t e d p u ls e compared w ith th e i n c i d e n t o r
re fe re n c e p u ls e .
T h is c o n d itio n h a s b een o b se rv e d i n th e p re v io u s
e v a lu a tio n o f r e s u l t s o f th e re fH .e c tio n e x p e rim e n ts .
I t was a n t i c i p a t e d
I
a f t e r p r e lim in a r y e x p e rim e n ts t h a t p u ls e s t r a n s m i t t e d th ro u g h s h o r t
s e c t i o n s , c u t - o f f w ith r e s p e c t to th e c a r r i e r fre q u e n c y o f th e i n c i d e n t
p u l s e , w ould be v e ry much d i s t o r t e d and t h a t com parison w ith
t h e o r e t i c a l l y p r e d i c t e d r e s u l t s w ould n o t be p o s s i b l e ,
HoY/ever, i t
was f e l t t h a t t h i s sh o u ld n o t p re c lu d e t h e i r i n c l u s i o n s in c e th e
p rim a ry i n t e r e s t o f t h i s w ork h a d c e n tr e d on th e tim e o f t r a n s m is s io n
o f e le c tr o m a g n e tic w aves th ro u g h a c u t - o f f w aveguide.
(S ee C h a p te r 1 ) .
F o r t h i s r e a s o n , t o o , th e i n t e r a c t i o n o f a p u ls e on tr a n s m is s io n
th ro u g h m o d ifie d s e c t i o n s , n o t c u t - o f f w ith r e s p e c t to th e c a r r i e r
fre q u e n c y , was s tu d ie d i n an a tte m p t to compare th e tr a n s m is s io n
i n t e r a c t i o n tim e s , o f o n ly s l i g h t l y d i s t o r t e d p u l s e s , w ith
t h e o r e t i c a l l y p r e d i c t e d tim e s .
As f o r th e r e f l e c t i o n e x p e rim e n ts
in v o lv in g lo n g g u id e s , r e c o r d in g s w ere a ls o made v fith an e x t r a 9oms
p a th le n g th t o o b t a i n some m easure o f th e a p p a re n t change i n c a r r i e r
fre q u e n c y o f th e em ergent p u l s e .
- 145"
( 4 . 12)
The r e c o r d in g p ro c e d u re was b r i e f l y a s f o l lo w s , w ith r e f e r e n c e to
f i g u r e 41 (14G ) , w hich shows th e system d ia g ra m m a tio a lly :
in c id e n t a t
th e p u ls e
was r e c o rd e d and r e - r e c o r d e d w ith an e x t r a 9oms lo n g
p l a i n s e c tio n in tr o d u c e d b e f o r e B.
T h is was c a r r i e d o u t f o r th e
u n m o d ified p u ls e and th e p u ls e t r a n s m i t t e d th ro u g h th e s h o r t s te p p e d
g u id e s e c t i o n in tr o d u c e d a t
F ig u re 42(147) shows th e co m p o site
r e c o rd in g f o r p u ls e s em erging from s e v e r a l s te p p e d g u id e o f l e n g th
3.2om s u n i t s f o r a c a r r i e r fre q u e n c y o f 6 . 075G-H2 .
4# 13
E v a lu a tio n o f th e t r a n s m i s s io n e x p e rim e n ts
I n g e n e r a l , i t p ro v ed much more d i f f i c u l t to o b t a i n c o n s i s t e n t
r e s u l t s w ith th e tr a n s m is s io n e x p e rim e n ts th a n \7 ith th e r e f l e c t i o n
e x p e rim e n ts .
T h is was b e c a u se f o r each r e c o r d in g , th e b r a s s i n s e r t s
h ad to be re a rim g e d and th e m o d ifie d gu id e r e p la c e d i n th e m easu rin g
arm o f th e m agic t e e se t-u p ^ t h i s in v o lv e d movement o f b o th th e
d e te c t i n g p a r t and th e t e s t - s e c t i o n .
T h is was th e g r e a t d is a d v a n ta g e
o f f i x i n g th e c a r r i e r fre q u e n c y and v a ry in g th e m o d ifie d s e c t i o n s .
F ig u re 4 3 ( ”148) g iv e s a t a b l e o f r e s u l t s o f th e m easured tim e o f t r a v e l
o v e r th e e x t r a 9oms p a th le n g t h f o r th e p u ls e s em erging from v a r io u s
m o d ifie d s e c t i o n s .
T h is shows t h e d e c re a s in g group t r a v e l tim e f o r
th e more d i s t o r t e d p u l s e s .
The com posite r e c o r d in g o f f i g u r e 42 shows
th e em ergent p u ls e to be more d i s t o r t e d , th e h i g h e r th e c u t - o f f
fre q u e n c y o f th e s e c t i o n .
T h is i s c o n s i s t e n t w ith th e t h e o r e t i c a l l y
p r e d ic te d c u rv e s o f th e m odulus o f th e tr a n s m is s io n c o e f f i c i e n t
fre q u e n c y o f f i g u r e 2}4(149),
F o r th e s e c tio n s w ith c u t - o f f f r e q u e n c ie s
much l e s s t h a n th e c a r r i e r fre q u e n c y o f th e p u l s e , v i z . f^ g = 4 * 3 8 ,
4*85&Hz , th e tr a n s m is s io n c o e f f i c i e n t i s n e a r l y u n i ty o v e r a l a r g e
ra n g e o f f r e q u e n c i e s n e a r t h e c a r r i e r fre q u e n c y f^ =
L ittle
d i s t o r t i o n i s e x p e c te d and t h i s i s shown i n th e r e l e v a n t r e c o r d in g s .
(4.13)
in p u t p u l s e '
m atched
lo a d
1 9cm8 "
p l a i n g uide
m atched
lo a d
W aveguide /
c o a x ia l
tr a n s f o r m e r
c o a x ia l
c ry s ta l
d e te c to r
sa m p lin g
o s c i ll o s c o p e
and
XT
re c o rd e r
i n p u t p u ls e
4—
9 cms. p l a i n g u id e
The m o d ifie d s e c t i o n was i n tr o d u c e d a t th e p o i n t A. A,
F ig u re
.
D iagram m atic R e p r e s e n ta tio n o f th e transm p^s^jnn
-147-
(4 .1 3 )
Tj ,
■jr # *
■
Ke**K,.a.awa.«».
'
aw
a m p litu d e
I
^ 1om 5
P u ls e
P ig u r e
.8 9
r e c o r d in g
lj2 »
n secs
6 . 075G*Hz
=
in d ic a t o r g iv e n
R e c o r d in g s o f t h e
o f th e
in
fig u r e
r e c tifie d
e n v e lo p e s
s t r a 3 .g h t th r o u g h p u l s e
th e p u ls e s tr a n s m itte d
o f m o d ifie d g u id e .
43
th r o u g h
(oC)
and
3 . Zorns
_
—1A}.8*
B road d im en sio n
i n cms.
(il-. 13)
Cut o f f fre q u e n c y P u ls e r e c o r d in g
f o r H . mode i n
in d ic a to r f o r
GH .
P ig . 42.
z
a
Croup t r a v e l
tim e o f p u ls e
o v e r 9 oms, i n
n seo s0
.43
3.485
4.3
3.285
4.58
3.085
4.85
P
-
2.885
5.2
7
.42
2.685
5.57
6
—
2.585
5.8
e
.40
2.485
6.05
2.385
6.3
C a r r i e r fre q u e n c y ,
.43
Ô
,41
= 6 «075GHz
P r e d i c t e d tim e o f t r a v e l o v e r 9 cms = ,425 n s e c s , f o r th e i n i t i a l
p u ls e
F ig u re 43#
T ab le o f M easurem ents, .s ta v in g th e re d u c e d b ro a d d im e n sio n s
f o r th e tr a n s m is s io n s e c tio n and th e group t r a v e l tim e s o f
th e i n c i d e n t and emerge)
—149*"
(2 .1 3 )
1 .0
.9
.8
.7
a
.6
Ic/A l
.5
.4
.3
.2
c u rv e f
i n GH
4 .5 8
4 .8 5
5 .2
5 .5 7
6 .0 5
6 .5 7
.1
fre q u e n c y
F ig u r e 4 4 .
i n GHz
Graph o f th e t h e o r e t i c a l l y p r e d i c t e d m odulus o f th e
'tr a n s m is s io n c o e f f i c i e n t v . fre q u e n c y f o r 6 m o d ifie d
s e c t i o n s . 3.2cms l o n g .
•'•1 ^0'-'
o f f ig u r e 4 2 ,
^ 3j
The te rm tr a n s m is s io n i n t e r a c t i o n d e la y tim e i s
in tr o d u c e d a g a in to d i s t i n g u i s h th e tr a n s m is s io n i n t e r a c t i o n tim e
m easured a t some d is ta n c e from th e b ack s u r f a c e o f th e s e c t i o n ,
from th e tr a n s m is s io n i n t e r a c t i o n tim e im m e d ia te ly a d ja c e n t to t h i s
su rfa c e *
As e x p e c te d th e tr a n s m is s io n i n t e r a c t i o n d e la y tim e s f o r
th e s e r e l a t i v e l y l i t t l e
d i s t o r t e d p u ls e s a g re e w ith th e p r e d i c t e d
v a lu e s ( f o r v a lu e s see f i g u r e
)).
I t w i l l be n o tic e d t h a t th e p e ak s o f th e much d i s t o r t e d p u ls e s
o c c u r b e fo r e th e p eak o f t h e u n d i s t o r t e d r e f e r e n c e p u l s e .
A p o s s ib le
e x p la n a tio n o f t h i s can be th o u g h t o f i n term s o f an in c r e a s e d
c a r r i e r fre q u e n c y .
T h is can be deduced from th e change i n gro u p
t r a v e l tim e shown i n f i g u r e
,
T h is w ould g iv e a l a r g e r g ro u p
v e l o c i t y i n th e u n m o d ified g u id e and a s m a lle r g ro u p t r a v e l tim e
compared w ith th e r e f e r e n c e p u ls e o v e r th e d i s t a n c e
p a r t i a l l y c o n t r i b u t e to th e e f f e c t *
T h is c o u ld
However, i t i s n o t p o s s i b l e to
d e s c rib e th e e f f e c t o f t h i s a p p a r e n tly i n c r e a s e d c a r r i e r fre q u e n c y
c o m p le te ly b e c a u se o f th e l a c k o f in f o r m a tio n a b o u t th e fre q u e n c y
sp ectrum o f th e i n c i d e n t p u lse *
The t r a n s m i s s io n e x p e rim e n ts w ere l i m i t e d to one s e t s in c e i t
was f e l t t h a t a s th e r e s u l t s c o u ld be e v a lu a te d o n ly i n a v e iy
q u a l i t a t i v e m anner, no f u r t h e r p u rp o se w ould be s e rv e d i n r e p e a t i n g
th e e x p e rim e n ts f o r o t h e r c a r r i e r f r e q u e n c ie s o r l e n g t h s o f m o d ifie d
guide*
-151
i l æ
( 4 .1 3 )
c u r v e T q2 i n
i
4 5a
G-Hz
6 .5 7
.k
R
in
tit
nsecB
i i l l
cu rve f
2
in
&Hz
R
in
nsecB
frequency
in
GHz
( l e n g t h o f s e c t i o n - 3 .2 cms.
F ig u r e s 4 5 a
tr a n s m is s io n
and b .
The g r a p h o f t h e
in te r a c tio n
tim e ,
t^ .
th e o r e tic a lly
)
p r e d ic te d
v e r s u s fr e q u e n c y .
-1 5 2 PART I I I ,
4 .1 4
(4.14)
The d i e l e c t r i c f i l l e d vm ve^uide T e s t S e c tio n R e f le c t i o n and T ra n sm iss io n e;
In tro d u c tio n
The seco n d microwave m ethod o f r e a l i z i n g p r a c t i c a l l y th e system
d e s c r ib e d t h e o r e t i c a l l y i n C h a p te r 3 i s fo u n d i n sandw iching a le n g th
o f u n f i l l e d g u id e b etw een two d i e l e c t r i c f i l l e d g u id e s .
T h is u n f i l l e d
g u id e h a s th e n , e f f e c t i v e l y , a h ig h e r c u t - o f f fre q u e n c y , f ^ g ,
compared w ith t h a t o f th e d i e l e c t r i c f i l l e d g u id e , f^^ and
j_
f
. = f
/( C
w here e
i s th e r e l a t i v e d i e l e c t r i c c o n s ta n t o f
c1
c2 ' ^ r^ ^
r
th e d i e l e c t r i c medium.
W ith th e ran g e o f c a r r i e r f r e q u e n c ie s o f
p u ls e s 3 .9 5 to 6.4G-Hz d e te rm in e d by th e t r a v e l l i n g wave tu b e ,
c a l c u l a t i o n s , u s in g th e t h e o r e t i c a l l y p r e d i c te d r e f l e c t i o n i n t e r a c t i o n
tim e s f o r a lo n g c u t - o f f s e c t i o n ( e x p r e s s io n ( 3 o 6 ,6 ) ) , i n d i c a t e d
p o s s ib le m ea su rab le tim e s u s in g p o ly s ty r e n e f i l l e d X band w aveg u id e.
The la u n c h in g s e c t i o n c o n s is te d o f a s h o r t l e n g t h o f Xband
w aveguide ( f ^ g = 6 , 56GHz) c o n ta in in g a movable s h o r t c i r c u i t and a
b lo c k o f p o l y s t y r e n e , a ls o mev-abl-e-.
The c r o s s - s e c t i o n a l d im en sio n s
o f t h i s b lo c k were t h a t o f th e X band w aveguide and i t was *35cms
lo n g i
A groove had b e e n d r i l l e d i n i t ,
th e b lo c k i t s e l f .
,7oms deep and as lo n g a s
The p ro b e w hich p e n e t r a t e d th e groove was i t s e l f
o n ly s l i g h t l y s m a lle r i n d ia m e te r (,3 o m s) th a n th e l e n g th o f th e
g ro o v e;
see f i g u r e s 4 6 a and 4 6 b (I5 3 ) .
The p ic k up s e c t i o n c o n s is te d
o f a s i m i l a r probe assem b ly b u t w ith th e probe r e s t i n g on th e s u r f a c e .
The re a s o n f o r t h i s w i l l become a p p a re n t a s th e e x p e rim e n ta l
p ro c e d u re i s e x p la in e d .
The s i g n a l p ic k e d up by th e p ro b e was
d e te c te d by a c o a x ia l c r y s t a l d io d e and th e d e te c t e d e n v e lo p e
d is p la y e d on th e e x t e r n a l l y t r i g g e r e d sam pling o s c illo s c o p e and
(4.14)
•153*
a) P la n
b ) S id e View
D
D ie le o trio
L
S m all d i e l e o t r i o b io o k
M M ovable s h o r t c i r c u i t
L au n ch in g p ro b e
G- Groove
P
P ic k -u p p ro b e
H
r
T e rm in a tio n
X
X b an d w aveguide
P robe h o l i e r
F ig u r e s l\£a. an d b
Laurichinp: s e c t i o n
p
90 cms
'30 cms
T
F ig u re h £ o m ' The com plete s e c t i o n
' F ig u re h S »
The d i e l e c t r i c f i l l e d w aveguide t e s t s e c t i o n f o r
r e f l e c t i o n m easurem ents
I 1,1
—13V"
re c o rd e d .
( Vu ^ 5)
The com plete t e s t s e c t i o n i s sho\m i n f i g u r e V 6c(l53 )*
The p o ly s ty r e n e f i l l i n g th e X hand v/aveguide was m ille d i n l e n g th s
o f a p p ro x im a te ly 30cms a s t h e r e was d i f f i c u l t y i n m a in ta in in g
t o le r a n c e s i n th e m il l in g o f p i e c e s lo n g e r t h a n t h i s .
The te r m in a t io n
a t T c o n s i s t e d o f e i t h e r a m ovable s h o r t c i r c u i t (w ith s h o r t le n g th s
o f p o ly s ty r e n e to m a in ta in i t s c o n ti n u i ty up to th e s h o r t c i r c u i t
fa c e ) o r a s e c t i o n o f u n f i l l e d g u id e , 32cms lo n g , fo llo w e d by 7,6oms
o f p o ly s ty r e n e f i l l e d g u id e and a m atched lo a d ( t h e l a t t e r c o m b in a tio n
w i l l be c a l l e d th e " c u t - o f f ” s e c t i o n f o r c o n v e n ie n c e ).
Vo16
E x p e rim e n ta l p ro c e d u re f o r th e r e f l e c t i o n e x p e rim e n ts .
I n g e n e r a l , th e method c o n s i s t e d o f th e fo llo w in g s t e p s :
f i r s t l y , th e l o c a t i o n o f a c o n v e n ie n t tim e z e ro on th e r e c o r d in g by
p la c in g a s h o r t c i r c u i t a t th e p la n e , T , ( f i g u r e V^) and o b s e rv in g
th e p u ls e r e f l e c t e d from t h i s ;
s e c o n d ly th e re p la c e m e n t o f th e
s h o r t c i r c u i t by th e " c u t - o f f “s e c t i o n and th e r e c o r d in g o f t h i s
second r e f l e c t e d p u l s e .
To o b t a i n a s u f f i c i e n t amount o f s ig n a l a t th e p i c k u p , la r g e
p e n e t r a t i o n o f th e la u n c h in g p ro b e w as n e c e s s a ry and th e c lo s e
p ro x im ity o f th e s h o r t c i r c u i t .
Movement o f th e s h o r t c i r c u i t away
from t h i s p ro b e ( w h i l s t s t i l l m a in ta in in g c o n ti n u i ty o f p o ly s ty r e n e
up to i t s f a c e ) made th e p u ls e i n c i d e n t a t P b ro a d e r and u n sy m m etrio al
on th e f a l l i n g ed g e, due t o th e s e p a r a tio n o f th e p u ls e r e f l e c t e d
from th e s h o r t c i r c u i t p a s t th e la u n c h in g p ro b e .
I t became
i n c r e a s i n g l y d i f f i c u l t a t th e h i g h e r c a r r i e r f r e q u e n c ie s to remove
e n t i r e l y t h i s unv/anted " t a i i l " s i g n a l .
Care was ta k e n , by m aldng
PT a s u i t a b l e le n g t h , to e n s u re t h a t th e r e f l e c t e d p u ls e to be
m easured d id n o t o v e rla p t h i s r e g io n o f unw anted s i g n a l .
-1 5 5 “
( 4 . 16)
I t was found t h a t p e n e t r a t i o n o f th e p ic k up p ro b e i n t o th e
p o ly s ty r e n e b lo c k l e d t o se v e re d i s t o r t i o n o f th e p u ls e p ro c e e d in g
p a s t P to w ard s th e te r m in a t io n so t h a t i t was a llo w e d to r e s t on th e
s u r f a c e o n ly .
Even so^ g r e a t c a re had to be ta k e n to e n su re t h a t
th e p ro b e d is tu r b e d th e p u ls e p ro c e e d in g to w a rd th e te r m in a t io n a s
little
as p o s s ib le .
The l a r g e s e p a r a t io n o f L and P e n s u re d t h a t any p u l s e s
r e f l e c t e d from th e p ic k up and th e n r e - r e f l e c t e d b ack from L to P
were s u f f i c i e n t l y rem oved i n t i n e so a s n o t to i n t e r f e r e w ith th e
p u ls e r e f l e c t e d from th e te rm in a tio n ^ T .
The d i s t a n c e , PT, was fo u n d a s p a r t o f th e e x p e rim e n ta l
te c h n iq u e and w i l l now be d e s c r ib e d ,
A c e r t a i n amount o f d i f f i c u l t y was found i n e v o lv in g a m easu rin g
te c h n iq u e , w h ic h , w ith o n ly s l i g h t a l t e r a t i o n , co u ld be u se d f o r a l l
c a r r i e r f r e q u e n c ie s .
The f o llo w in g c r i t e r i a were made i n lo o k in g f o r
a s u c c e s s f u l te c h n iq u e ,
P i r s t l y , th e form o f th e m e a su rin g system
n e c e s s a r i l y in v o lv e d th e d is p la y and r e c o r d in g o f b o th th e i n c i d e n t
and r e f l e c t e d p u ls e s t o g e t h e r .
T h e ir m u tu al i n t e r f e r e n c e was re d u c e d
to a minimum by e n la r g in g th e d is ta n c e ^ FT, so t h a t t h e i r o v e rla p
was s m a ll.
S eco n d ly , on th e o t h e r h a n d , i t was d e s i r a b l e to keep t h i s
d is ta n c e a s sm a ll as p o s s i b l e so a s to make m easurem ents o f th e
r e f l e c t i o n i n t e r a c t i o n tim e a s n e a r to th e f r o n t o f th e c u t - o f f
s e c tio n as p o s s ib le .
T h ir d l y , i t was a ls o d e s i r a b l e to keep th e
d is ta n c e FT sm a ll so t h a t t h e r e were a s fev7 as p o s s ib l e p o ly s ty r e n e
b lo c k s betw een P and T,
Any u n e v en n e ss o f th e m il l in g o f th e ends
o f th e b lo c k s le a d s to d i s c o n t i n u i t i e s o f th e p o ly s ty re n e w ith p o s s ib le
unw anted r e f l e c t i o n s .
-1 5 6 -
( 4 .1 6 )
The minimmi d is ta n c e betw een P and T was a p p ro x im a te ly 30eras,
F o r each c a r r i e r fre q u e n c y o f th e i n c i d e n t p u l s e , p u ls e s r e f l e c t e d
from a few s h o r t c i r c u i t p o s i t i o n s , s e p a r a te d by sm a ll Icnown d i s t a n c e s ,
were re c o r d e d .
T hese sm a ll d i s t a n c e s w ere chosen so t h a t th e e x tr a
p a th l e n g th in tr o d u c e d by them gave a tim e d e la y o f th e same o r d e r
a s th e r e f l e c t i o n i n t e r a c t i o n tim e s w hich m ight be e x p e c te d f o r th e
" c u t-o ff" s e c tio n s .
The d i s t a n c e o f th e f i r s t s h o r t c i r c u i t p o s i t i o n
from T was v a r i e d u n t i l th e r e c o r d in g s showed p u ls e s o f e q u a l p eak
h e ig h t s and p a i r s o f c o rre s p o n d in g p o i n ts on a d ja c e n t p u l s e s s e p a r a te d
b y th e same d i s t a n c e .
When such a p o s i t i o n o r a c lo s e a p p ro x im a tio n
to i t had b e e n fo u n d , i t was c o n s id e re d s u i t a b l y f r e e from unw anted
r e f l e c t i o n s to be u s e f u l f o r th e r e c o rd in g o f p u ls e s r e f l e c t e d from
th e c u t - o f f s e c t i o n s ,
F ig u r e s 4 7 ( 57) and 4 8 (1 ^ 8 ) show r e c o r d in g s o f p u ls e s w ith
c a r r i e r f r e q u e n c ie s 6 ,2 5 , 6 ,0 7 6 and 6,33G-Hz,, r e s p e c t i v e l y .
S h o rt
e x p la n a to ry n o te s a re in c lu d e d on them b u t a few comments w i l l be
made h e r e .
F i r s t l y , f o r th e lo w e s t c a r r i e r fre q u e n c y p u l s e , f i g u r e
48a shows an a lm o st sy m m e tric a l r e c o rd in g o f th e i n c i d e n t p u ls e b u t
w ith a "bump" n e a r th e b a se l i n e o f th e f a l l i n g ed g e.
How ever, th e
tvfo s h o r t c i r c u i t r e c o r d in g s d id n o t o v e rla p t h i s r e g io n and form ed a
good p a i r .
F o r th e c a r r i e r f r e q u e n c y , f ^ = 6 , 25GHz, i t v f ill be se e n i n
f i g u r e 47 t h a t th e "bump" on th e f a l l i n g edge o f th e i n c i d e n t p u l s e ,
o c c u rs n e a r e r th e p eak o f th e p u l s e .
However, th e p u l s e s r e f l e c t e d
from th e s h o r t c i r c u i t p o s i t i o n s form a sy m m etrica l p a i r .
F ig u re 48b f o r f ^ = 6,325GHz shows a v e iy b ro a d e n e d i n c i d e n t
p u l s e , p ro b a b ly due to t h e unw anted "bump" b e in g v e ry n e a r th e p eak
-157-
(4 .1 6 )
J„.U
a m p litu d e
vtim©
.93
1cm s
n secs,
6.25
a
=
a and
0
-
p u ls e s
b
«• p u l s e
d
-
P ig u r e
24.7 .
r e fle c te d
fr o m
r e f l e c t e d fr o m
GHz
6 . 56GHz
sh o rt c ir c u it
” c u t - o f f ” w a v e g u id e
in c id e n t p u ls e
R e c o r d in g s o f t h e
ta k e n
fille d
d u r in g
r e c tifie d
r e fle c tio n
w a v e g u id e t e s t
e n v e lo p e s o f p u ls e s
e x p e r im e n ts w it h
s e c tio n .
d ie le c tr ic
158-
(4.16)
= 6.075&H5
t(
\
I.
j.! fi
i
II')
' ■ Il l i i . I ! t ■i 1h n i I i ! 1
f e b . 325GHz
II I I Ii I I I
if8b
I
t
i
ïsàycnyi:
wLmUwïil^enÀi HÉ
i h t i- iH4IH4 li^iWii:
a m ^ i tu d e
•—^ c m
For a ,
s
b,
F i g u r e if-G.
,9 5
0
n seos.
o2
and d ~ se e
fig u r e
R e c o r d in g s o f t h e
p u ls e s
w ith
ta k e n
th e
6 .5 6 G H z
if 7
r e c tifie d
d u r in g
d ie le c tr ic
e n v e lo p e s o f
r e fle c tio n
fille d
e x p e r im e n ts
w a v e g u id e
te st
s e c tio n .
-1 5 9 on th e f a l l i n g ed g e.
(4 .1 7 )
The l a s t p a i r o f s h o rt c i r c u i t r e c o r d i n g s ,
a and c , more n e a r ly a p p ro a c h th e i d e a l c o n d itio n s and so t h i s p o s i t i o n
was u se d f o r r e c o r d in g s a s shovm above i t *
4 .1 7
E v a lu a tio n o f th e m ea su rin g system and r e s u l t s o b t a i n e d .
T here w ere s e v e r a l d is a d v a n ta g e s , com pared w ith th e w aveguide
WG-IZf. sy stem o f P a r t I I , i n talcin g m easurem ents o f th e r e f l e c t i o n
i n t e r a c t i o n tim e s i n t h i s d i e l e c t r i c f i l l e d X b and t e s t s e c t i o n .
However, i t s o u ts ta n d in g a d v a n ta g e waa t h a t i t r e a l i z e d more
r i g o r o u s l y i n p r a c t i c a l form th e system o f f i g u r e 2b and a n a ly z e d
t h e o r e t i c a l l y i n C h a p te r 3 - t h a t i s to sa y t h a t th e one mode o n ly
a n a l y s i s u s e d i s s t r i c t l y a p p lic a b le f o r th e d i s c o n t i n u i t y a c ro s s
th e d i e l e c t r i c / a i r b o u n d a r ie s .
F i r s t l y , i t p ro v e d d i f f i c u l t to e s t a b l i s h a h ig h enough
s i g n a l l e v e l a t th e d e t e c t o r f o r s u i t a b l e d is p l a y on and re c o r d in g
from th e sam p lin g o s c i ll o s c o p e w hich was s u s c e p t ib l e to a f a i r
d eg ree o f p ic k - u p .
th e p ic k -u p system*
The m ain so u rc e o f d i f f i c u l t y i n t h i s c a se was
I t h a s p r e v io u s ly b een p o in te d o u t t h a t th e
p ic k up p ro b e r e s t e d on th e s u r f a c e o f th e p o ly s ty re n e so t h a t i t
i n t e r f e r e d n e g l i g i b l y w i t h t h e p u ls e p ro c e e d in g p a s t i t to w a rd th e
t e r m in a t io n .
s ig n a l;
T h is r e s u l t e d i n th e d e t e c t i o n o f a sm a ll amount o f
f o r t h i s r e a s o n , th e la u n c h in g p ro b e was a llo w e d to p e n e t r a t e
th e p o ly s ty r e n e a lm o st to i t s f u l l d e p th .
S e c o n d ly , w ith t h i s p ro b in g system w ith s im u lta n e o u s d is p la y
o f th e i n c i d e n t and r e f l e c t e d p u l s e s , i t was n e c e s s a r y t o p o s i t i o n
th e p ro b e some d is ta n c e away from th e f r o n t s u rfa c e o f th e c u t - o f f
s e c tip n to re d u c e th e m u tu a l i n t e r f e r e n c e o f th e p u l s e s .
T hus, i t
was n o t p o s s i b l e to m easure th e r e f l e c t i o n i n t e r a c t i o n tim e a d ja c e n t
"•160‘”
(^{-«17)
to th e c u t - o f f g u id e and a s p r e v io u s ly , th e m easured tim e h a s b een
d e s ig n a te d a s th e " r e f l e c t i o n i n t e r a c t i o n d e la y tim e " .
How ever, i n
th e s te p p e d w aveguide r e f l e c t i o n m easurem ents, i t was n o t p o s s ib le
to m easure a d ja c e n t to th e c u t - o f f g u id e f o r a s i m i l a r r e a s o n .
T h ir d l y , w h i l s t i t h a d b e en p o s s ib l e to u se i s o l a t o r s and
m atched lo a d s i n th e WG14- system to re d u c e r e f l e c t i o n s , t h i s was n o t
so i n th e p o ly s ty r e n e f i l l e d X band t e s t s e c tio n ,
A l o s s y lo a d
w ith ro u g h ly m atching p r o p e r t i e s was u se d b u t f o r th e m ost p a r t ,
i t v/as n e c e s s a r y to i d e n t i f y th e r e f l e c t i o n s and a d j u s t th e d is ta n c e
FT so t h a t th e r e f l e c t e d p u ls e from T was n o t d e te c te d i n p o s i t i o n s
o f unw anted r e f l e c t i o n s .
T h is was d e s c r ib e d i n s e c tio n (/|-e l6 ).
Now, assum ing t h a t th e b e s t p o s s ib l e m easurem ents have b e en
made, th e r e f l e c t i o n i n t e r a c t i o n d e la y tim e s f o r th e d i f f e r e n t
c a r r i e r f r e q u e n c ie s have b e en p l o t t e d i n f i g u r e 2j.9(l61 ) • The
p ro c e d u re f o r d e d u cin g th e s e tim e s from th e p u ls e r e c o r d in g s h a s
b e en d e s c r ib e d i n S e c tio n A#7 o f P a r t I I ,
The t h e o r e t i c a l l y
p r e d i c t e d r e f l e c t i o n i n t e r a c t i o n tim e s have a ls o b e en p l o t t e d .
It
w i l l be se e n t h a t as th e r a t i o o f c a r r i e r fre q u e n c y to c u t - o f f
fre q u e n c y o f th e u n f i l l e d s e c t i o n , a p p ro a c h e s u n i t y , th e m easured
d e la y tim e i n c r e a s e s , a s does th e t h e o r e t i c a l l y p r e d i c t e d o n e ,
For
th e lo w e r v a lu e s , th e m easu red tim e s a g re e w ith th e t h e o r e t i c a l l y
p r e d i c t e d o n e s.
F o r c a r r i e r f r e q u e n c ie s n e a r f ^ g , th e m easured
d e la y tim e i s g r e a t e r th a n th e p r e d i c t e d v a lu e .
T h is i s i n
a c c o rd a n c e w ith th e t r e n d f o r th e s te p p e d w aveguide r e f l e c t i o n
•1é1
(4.17)
in
n seo s
6*1
6.2
6.3
6.4
fr e q u e n c y i n G-H^
f
J
M easured d e la y tim e
—
P r e d i c t e d cu rv e
p
^
P ig u r e .49,
c u t o f f fr e q u e n c y o f u n f i l l e d s e c t i o n
= 6.36GHz
The g ra p h o f r e f l e c t i o n i n t e r a c t i o n tim e , t ^ . v e rs u s
fre q u e n c y f o r t h e d i e l e c t r i c f i l l e d w aveguide t e s t
s e c tio n .
•“'162—
e x p e rim e n ts ( s e c t i o n (4 * 9 ))*
(4'* ' /}
F o llo w in g th e r e a s o n in g g iv e n t h e r e ,
i t can he assum ed t h a t a s th e c a r r i e r fre q u e n c y a p p ro a c h e s th e
c u t - o f f f re q u e n c y , f ^ g , more c l o s e l y , more and more o f th e fre q u e n c y
sp e ctru m o f th e i n c i d e n t p u ls e
T d .ll
l i e above c u t - o f f and t h a t th e
r e f l e c t e d p u ls e v f i ll become more and more d i s t o r t e d .
T h is i s
d e m o n s tra te d i n f i g u r e 4-7(137) and 4-8(1 ^8 ) where a co m p ariso n o f
th e p u ls e s r e f l e c t e d from th e s h o r t c i r c u i t , ( a ) , and c u t - o f f
w av eg u id e, ( b ) , show them becom ing more d i s s i m i l a r i n h e ig h t f o r
th e h ig h e r c a r r i e r f re q u e n c y .
T h e r e f o r e , a d is c re p a n c y betw een th e
m easu red tim e and th e tim e p r e d i c t e d t h e o r e t i c a l l y from a knowledge
o f th e c a r r i e r fre q u e n c y a lo n e , i s to
be e x p e c te d .
F u r t h e r e x p e rim e n ts to g a in some i n d i c a t i o n o f any change i n
group t r a v e l tim e b ack to th e p ro b e from th e t e r m in a tio n due to
t h i s d i s t o r t i o n were n o t made a s i n P a r t I I ,
4 ,1 8
T ra n s m is s io n e x p e rim e n ts w ith s h o r t u n f i l l e d s e c tio n s
H e re , a g a in , a s w ith th e tr a n s m is s io n e x p e rim e n ts f o r th e s te p p e d
w aveguide WG-14- c o n f i g u r a ti o n , b eca u se o f th e l i m i t e d in f o r m a tio n
ab o u t th e i n c i d e n t p u ls e and th e d i s t o r t i o n o f th e em ergent p u l s e , a
q u a n t i t a t i v e a sse ss m e n t o f th e m easurem ents w i l l n o t be p o s s i b l e .
However, th e y a re in c lu d e d f o r th e sake o f c o m p le te n e s s, w ith an
a tte m p t to a s s e s s any change i n group t r a v e l tim e o f th e em ergent
p u lse *
4-019
E x p e rim e n ta l P ro c e d u re
The a p p a r a tu s was b a s i c a l l y t h a t o f f i g u r e 4 ^ (1 5 3 ), w ith an
a b s o rb in g s e c t i o n a t th e t e r m in a t io n , T,
I n f i g u r e 30(163)
,
Xg, X^, w ere b lo c k s o f p o ly s ty r e n e o f e q u a l l e n g t h , x ,a n d C was th e
s h o r t l e n g t h , 1*3 cms lo n g , o f u n f i l l e d g u id e - th e " c u t - o f f "
-163-
(4.19)
P
l o s s y medium
b)
F ig u re 3 0 •
Diagram o f th e t e s t s e c t i o n f o r th e
tr a n s m is s io n e x p e rim e n ts .
~ l6 k ^
s e c tio n .
( 4 . 19)
The tim e o f t r a v e l o f th e o r i g i n a l p u ls e th ro u g h th e d i s t a n c e ,
X, i s d e s ig n a te d by t ^ ;
th e tim e o f t r a v e l f o r d is ta n c e x o f th e
t
p u ls e t r a n s m i t t e d th ro u g h th e " c u t - o f f " s e c t i o n by t ^ and i t s
tr a n s m is s io n tim e , t^*
W ith o u t th e c u t - o f f s e c t i o n betw een th e la u n c h in g and p i c k up
p o i n ts th e ...tine. Of t r a v e l i s g iv e n by
T,
= 3 t^
( 4 . 1 9 . 1)
F o r th e system i l l u s t r a t e d i n f i g u r e 50'b(l53) th e t o t a l tim e o f
tra v e l is
T
c
= t
O
+ t . + 2t*
u
(Zi.,19*2)
O
and f o r t h a t i n f i g u r e 5 0 c (l6 3 )
is
T? = 2 t + t , + t *
3
o
t
o
. . . ( 4 . 1 9 . 3)
The d i f f e r e n c e (T ^ - T^) y i e l d s th e d i f f e r e n c e i n gro u p t r a v e l
tim e s o v e r x cms o f th e i n c i d e n t and em ergent p u l s e s .
(T^ - T^) +
(T^ - T^) y i e l d s th e tr a n s m is s io n tim e o f th e p u ls e th ro u g h th e
c u t - o f f g u id e .
f i r s t l y , th e i n c i d e n t p u ls e was r e c o rd e d w ith o u t th e c u t - o f f
s e c t i o n betw een L and P .
was p la c e d a t
T hen, th e u n f i l l e d s e c t i o n o f g u id e , C,
and th e p o ly s ty r e n e b lo c k ,
, p u l l e d alo n g u n t i l
th e f a c e ..o r i g i n a l l y a t L^^w&s th e n a t th e p o s i t i o n ,
, le a v in g a
s e c t i o n o f u n f i l l e d g u id e , I # , b e tw ee n th e b l o c k s , X^ and X^ - see
f i g u r e 3 0 b ( l6 3 ) .
The p u ls e t r a n s m i t t e d th ro u g h th e c u t - o f f g u id e
was r e c o rd e d .
The S e c tio n , C, was rem oved from
and th e g a p ,I # , c lo s e d u p ,
C was th e n i n tr o d u c e d a t R and th e fa c e o f th e b lo c k X , e a se d fo rw a rd
o
3
u n t i l i t was c o in c id e n t w ith R.
fG- b etw een X^ and X^.
T h is l e f t a s e c t i o n o f u n f i l l e d g u id e
~
4 .2 0
( 4 . 20)
165 ~
E v a lu a tio n o f th e mea s u r i ng s y s te m, and t h e r e o o rd ln ^ s o f th e
t r a n s m itte d pul se s
F ig u r e s 5 'la (l6 6 ) and 5 1 b (l6 6 ) show re c o r d in g s o f th e r e c t i f i e d
e n v e lo p e s o f th e u n m o d ifie d and t r a n s m i t t e d p u ls e s f o r c a r r i e r
f r e q u e n c ie s o f 5»95G-Hz and 6.3G-Hz.
F o r th e p u ls e o f h i g h e r c a r r i e r
fr e q u e n c y , th e unw anted " t a i l " on th e f a l l i n g edge i s p r e s e n t a s i n
th e r e f l e c t i o n e x p e rim e n ts .
The p u l s e s t r a n s m i t t e d th ro u g h th e
u n f i l l e d s e c t i o n a r e re d u c e d i n s i z e and have v e iy f l a t t e n e d p e a k s ,
w hich made e s tim a tio n s o f th e tim e p o s i t i o n s o f th e peaks e x tre m e ly
d iffic u lt.
T h e re fo re t h i s p a r t o f th e e x p e rim e n t f a i l e d t o y i e l d
r e s u l t s o f changes i n g roup t r a v e l tim e s .
T h is p a r t o f th e
e x p e rim e n ts vms th e m ost d i f f i c u l t t o p e rfo rm b e c a u se f o r each
r e c o r d in g , th e a p p a r a tu s a s a w hole was more o r l e s s d is m a n tle d and
th e n re a s s e m b le d .
It
T d .ll
be rem embered t h a t f o r th e r e f l e c t i o n
e x p e rim e n ts t h a t though th e te r m in a tio n a t T (s e e f i g u r e 4 ^ (1 5 3 ))was
changed c o n ti n u a l l y th ro u g h th e c o u rse o f th e e x p e rim e n t, th e
la u n c h in g and p ic k -u p s e c t i o n s rem ained unmoved.
The f r e q u e n t
movement o f a p p a ra tu s f o r th e tr a n s m is s io n e x p e rim e n ts d id n o t make
f o r c o n s is te n c y o f r e s u l t s .
The b a s i c so u rc e o f d i f f i c u l t y was th e
n eed f o r c o n t i n u i t y o f th e p o ly s ty r e n e f i l l i n g th e X band w aveguide.
To make e s t im a t e s o f ch an g es i n group t r a v e l tim e , i t was n e c e s s a iy
to in tr o d u c e th e u n f i l l e d , c u t - o f f s e c t i o n a t two d i f f e r e n t p o i n ts
betw een th e la u n c h in g and p ic k - u p s e c t i o n s .
When th e e x p e rim e n ts
were p e rfo rm e d p l a i n g u id e s e c t i o n s th e same l e n g th a s th e
p o ly s ty r e n e b lo c k s X^, X^ an d X^ were n o t a v a i l a b l e .
B ecause o f
t h i s , th e p ro c e d u re o f s e c t i o n (4 .1 9 ) was a d o p te d i n w hich s m a ll
movements o f b lo c k s X^ and X^ w ere made.
E v eiy e f f o r t was made t o
m a in ta in c o n t i n u i t y o f th e p o ly s ty r e n e b u t when t h i s was n o t
'166'
51a
f
( 4 .2 0 )
= 6.30Hz
= 5.950Hz
J-l
•>
Kimtsim
a m p litu d e
^
tim e
1
cm =
i « in itia l
~C -
to
co rresp o n d to
F ig u r e
51
2
o 6 .5 6
GHz
p u ls e
tr a n s m itte d p u ls e
The tr a n s m itte d
co rresp on d
f
#95 n seo o n d s
tmrm
p u ls e
th e
in
th e
u p p er r e c o r d in g s
c o n fig u r a tio n o f
fig u r e
fig u r e
o f fig u r e s
5 0 o w h ile
th e
^1a)
and b)
lo w e r o n e s
50b,
R e c o r d in g s o f t h e
and tr a n s m itte d
r e c tifie d
p u ls e ,
e n v e lo p e s o f
ta k e n w ith
f i l l e d waveguide te s t seotion.
th e
th e
in itia l
d ie le o tr io
rjJKS
-1 6 7 -
(4 .2 0 )
a c h ie v e d , a m a jo r so u rc e o f e r r o r was p ro d u c e d .
4.21
C oncluding rem arks
The tr a n s m is s io n e x p e rim e n ts o f th e p re o e e d in g s e c tio n s
com plete th e p r a c t i c a l work u n d e rta k e n and d e s c r ib e d i n t h i s
c h a p te r .
E v a lu a tio n o f th e r e s u i t s o b ta in e d h a s b een made i n
s e c tio n s su c c e e d in g th e e x p e rim e n ta l p ro c e d u re f o r o b ta in in g
them .
The a u th o r f e l t t h a t t h i s was th e b e s t c o u rse s in c e
i n t e r p r e t a t i o n o f th e r e s u l t s c o u ld o n ly be made i n th e l i g h t
o f th e m ethod o f o b ta in in g them .
F u r t h e r d is c u s s io n o f th e
e x p e rim e n ts and r e s u l t s , a s a whole^ i s r e s e r v e d f o r th e f i n a l
c h a p te r .
“ i 68~
CHAPTER 5 .
C o n c lu s io n s ,
5 .0
I n t r o d u c t io n
The aim o f t h i s i n v e s t i g a t i o n h a s b een to make a tim e
d ep en d e n t stu d y o f th e waves p r o p a g a tin g i n a w aveguide o r
s i m i l a r d i s p e r s i v e medium ( f o r exam ple, p la s m a ), whose f r e q u e n c ie s
a re i n th e s o - c a l l e d c u t - o f f ra n g e .
The v e iy n a tu r e o f th e problem
h a s d iv id e d th e i n v e s t i g a t i o n i n to two p a r t s , th e f i r s t d e s c r ib e d i n
C h a p te r 2 and th e se c o n d i n C h a p te rs 3 and 4 ,
The f i r s t p a r t h a s b een c o n ce rn e d w ith th e e v a n e s c e n t wave i n
an u n te rm in a te d g u id e , i n w hich th e tim e av erag e P o y n tin g v e c t o r i s
z ero f o r s te a d y s t a t e c o n d itio n s .
W ith th e i n t r o d u c t i o n o f a d e te c t i n g d e v ic e ( f o r exam ple, a
s e c t i o n o f p r o p a g a tin g g u id e a t a p o i n t i n th e c u t - o f f g u id e ) i t
i s no lo n g e r p o s s ib l e to c o n s id e r th e wave i n th e fo rw a rd d i r e c t i o n
a lo n e , a s i n th e p r o p a g a tin g c a s e .
r e f l e c t e d waves m ust b e su p e rp o se d .
The fo rw a rd and backw ard
T h is s u p e r p o s i t io n g iv e s r i s e to
a n o n -z e ro tim e a v e ra g e P o y n tin g v e c t o r i n th e fo rv fard d i r e c t i o n .
I t i s t h i s d i s t i n c t d i f f e r e n c e t h a t h a s l e d to th e d i v i s i o n o f th e
i n v e s t i g a t i o n . i n t o two p a r t s .
S in ce t h e r e i s a d e te c t a b l e power
flo w to th e t e r m in a t io n , th e e x p e rim e n ta l i n v e s t i g a t i o n s d e s c r ib e d
i n C h a p te r 2^ were p e rfo rm e d to v e r i f y some o f th e t h e o r e t i c a l
p r e d i c t i o n s o f C h a p te r 3 .
B ecause o f th e d i f f e r e n c e , no com parison w i l l be made o f th e
two s i t u a t i o n s and t h i s f i n a l c h a p te r i s a ls o d iv id e d i n t o two
p a rts .
S e c tio n ( 5 .1 ) w i l l d e a l b r i e f l y w ith th e a n a l y s i s and
-1 6 9 -
(5 .0 )
i l l u s t r a t i o n s p r e s e n te d i n C h a p te r 2 w h ile s e c t i o n ( 5 .2 ) w i l l d e a l
w ith th e a n a l y s i s and t h e o r e t i c a l p r e d i c t i o n s o f C h a p te r 3 w ith th e
oom plem entaiy e x p e rim e n ta l i n v e s t i g a t i o n and r e s u l t s o f C h a p te r 4#
3*1
The tim e d e p e n d e n t s tu d y o f t h e e v a n e sc e n t w ave.
U sing a L a p la c e tra n s fo r m te c h n iq u e , th e a p p l i c a t i o n o f a
su d d en e le c tr o m a g n e tic d is tu r b a n c e a t z = o i n an u n te rm in a te d
w aveguide o r p lasm a h a s been s tu d ie d i n d e t a i l .
The r e s t r i c t i o n s
have b een i n tr o d u c e d , w hich must be p la c e d upon th e f i e l d com ponents
a t th e w a v e fro n t ( t = z /o ) i n o r d e r t h a t th e s o l u t i o n s be e l e c t r o ­
m ag n e tic ( t h a t i s , s a t i s f y M axw ell*s e q u a tio n s ) .
T hese r e s t r i c t i o n s ,
i n t u r n , have l e d to r e s t r i c t i o n s on th e tra n s fo rm ( w ith r e s p e c t to
th e com plex f r e q u e n c y , s) o f th e i n i t i a l c o n d itio n i n th e tim e dom ain
( t h a t i s , th e e x c i t a t i o n ) a t z = o and u l t i m a t e l y on th e i n i t i a l
c o n d itio n i t s e l f .
S e v e ra l a u th o r s have u se d t h i s tr a n s fo rm te c h n iq u e to c o n s id e r
th e a p p l i c a t i o n o f a u n i t s t e p m o d u lated c a r r i e r wave to a w aveguide
o r p lasm a , c u t - o f f o r p r o p a g a tin g w ith r e s p e c t t o th e c a r r i e r
fre q u e n c y .
W h ils t m ost a u th o r s have c o n fin e d th e m se lv e s to a s tu d y
o f one component o f th e e le c tr o m a g n e tic f i e l d o n ly , G e rrillo ^ ^ ^ ^ and
R ubinow icz^^^^ hav e c o n s id e re d a l l th e com ponents o f th e f i e l d f o r
f^ > f ^ .
How ever, i n t h i s i n v e s t i g a t i o n , R u b in o w icz*s m ethod h a s
b een s u c c e s s f u ll y e x te n d e d to th e c a s e , f ^ < f ^ .
i t s r e s u l t s have b e e n p r e s e n te d i n C h a p te r 2.
T h is a n a l y s i s and
I t w i l l be b r i e f l y
re v ie w e d h e r e .
I t was d e c id e d to s tu d y th e b e h a v io u r o f a p a i r o f t r a n s v e r s e
com ponents i n o r d e r to e x te n d th e scope o f th e a n a l y s i s to i n c l u d e
p r o p a g a tio n o f p la n e w aves i n a p la sm a .
A t a p o i n t i n th e p la sm a ,
(5.1 J
w17Qw
th e form o f one o f th e s e t r a n s v e r s e com ponents was s p e c i f i e d a s th e
u n i t s te p m o d u lated c a r r i e r w ave,
f ( o ,t)
=
o
t< o
f ( o ,t)
=
l(t)
s in ( (jü^t)
a l l o t h e r com ponents b e in g z e r o .
t> o ,
I t m i l be s t r e s s e d t h a t t h i s i s
th e form o f th e e x c i t a t i o n ta k e n a s th e i n i t i a l c o n d itio n i n th e tim e
dom ain
th e c u t - o f f gu id e o r p la sm a .
T here i s no r e f e r e n c e to th e
la u n c h in g o f t h i s s i g n a l i n t o th e d i s p e r s i v e medium.
U sing a p a r t i c u l a r i n t e g r a t i o n p a th i n th e com plex fre q u e n c y
p l a n e , th e i n v e r s i o n i n t e g r a l s f o r a p a i r o f t r a n s v e r s e f i e l d com ponents
have b e e n e v a lu a te d e x a c tly a s a f u n c t i o n o f th e tim e , t , and d i s t a n c e ,
2 , from th e o r i g i n .
F o r g iv e n v a lu e s o f th e p a r a m e te r s , z /X ^ and f ^ / f ^ , n u m e ric a l
exam ples have b een com puted a s a f u n c t i o n o f th e n o rm a liz e d tim e , oô^t.
The e x a c t and ap p ro x im ate s o l u t io n s have b e en com pared.
I t was shown
t h a t f o r la r g e tim e , th e ap p ro x im ate s o l u t i o n c o u ld b e u se d to r e p r e s e n t
a c c u r a t e l y th e e x a c t o n e .
However, i t was n o t p o s s ib l e to d e fin e th e
lo w e r l i m i t o f v a l i d i t y f o r th e a p p ro x im a te s o l u t i o n s .
F o r v e iy l a r g e
tim e ( t •5 > z/c), th e t r a n s v e r s e f i e l d com ponents re a c h th e s te a d y s t a t e
b u t i n t h i s a s p e c t a n o t ic e a b l e d i f f e r e n c e b etw een th e two com ponents
a ris e s .
The s te a d y s t a t e i s re a c h e d w i t h in s e v e r a l c y c le s o f th e
c a r r i e r wave f o r th e t r a n s v e r s e m a g n e t i c / e l e c t r i c f i e l d component
whose form i s s p e c i f i e d a t z = o , ( t h i s i s , f o r th e v a lu e s z /
i n th e n u m e ric a l e x a m p le s).
u se d
T h is i s n o t so f o r th e d e r iv e d t r a n s v e r s e
\
e le c tric /m a g n e tic f i e l d .
The tim e ta k e n f o r t h i s d e r iv e d component to
re a c h ste a d y s t a t e may be m easured i n t e n s o f c y c le s o f th e c a r r i e r
w ave.
I t h a s b e e n s u g g e s te d t h a t t h i s d i f f e r e n c e i s due to th e
«-1 Yi —
( 5 *1)
r e s t r i c t e d n a tu r e o f th e e x c i t a t i o n o f th e f i e l d a t 2 = o i n th e
d i s p e r s i v e medium:
i t was assum ed t h a t th e r e was o n ly one non^^zero
component a t z = o .
P re v io u s a u th o r s , f o r th e c a s e , f
^
' 0
> f , have co n cern ed
c'
th e m se lv e s w ith th e b e h a v io u r o f a p a r t i c u l a r component a s a f u n c t i o n
o f tim e f o r a g iv e n d i s t a n c e , w ith a view to m e a su rin g th e d e g r a d a tio n
o f th e s i g n a l on p a s s a g e th ro u g h th e d i s p e r s iv e medium.
However, f o r
f ^ < f ^ i n th e u n te rm in a te d medium, w h i l s t i n i t i a l l y a wave i s p r o p a g a te d
i n i t , th e pow er floY/, a s g iv e n by th e tim e a v e ra g e P o y n tin g v e c t o r , i s
z e ro f o r th e s te a d y s t a t e .
I n view o f t h i s p r o p e r t y , th e s tu d y o f th e
b e h a v io u r o f th e in s ta n ta n e o u s P o y n tin g v e c t o r i s v e iy i n t e r e s t i n g .
I n g e n e r a l, f o r a tim e a f t e r th e a r r i v a l o f th e w a v e fro n t t h e r e a r e
p o s i t i v e and s m a lle r n e g a tiv e e x c u r s io n s o f th e P o y n tin g v e c t o r .
U n til
t h e r e i s a maximum p e ak v a lu e f o r th e p o s i t i v e e x c u rs io n s i t can b e
s a i d t h a t t h e r e i s a n e t p o s i t i v e en erg y f l u x i n th e fo rw a rd d i r e c t i o n
i n th e medium.
A f t e r t h i s tim e , i t i s d i f f i c u l t to p o in t o u t a
d e f i n i t e t r e n d th o u g h i n p a r t i c u l a r c a s e s , t h e r e i s a ten d e n cy to w a rd s
t h e e q u a l i t y o f th e p o s i t i v e and n e g a tiv e e x c u r s io n s and th e tim e th e y
o ccu p y.
F o r l a r g e r tim e , when th e ste a d y s t a t e i s e s t a b l i s h e d , t h e r e
w i l l be tv7o e q u a l p o s i t i v e and n e g a tiv e e x c u rs io n s o f th e P o y n tin g
v e c t o r p e r c y c le o f th e
c a r r i e r wave.
A f u r t h e r s h i f t o f em p h asis was made
i n c o n s id e r in g
th e
v a r i a t i o n o f th e t r a n s v e r s e f i e l d com ponents a s a f u n c t i o n o f d i s t a n c e
a lo n g th e g u id e , f o r a g iv e n tim e .
F o r th e s te a d y s t a t e , a l l p o i n ts
i n t h i s medium o f im a g in a ry c h a r a c t e r i s t i c im pedance o s c i l l a t e i n
p h ase.
The g ra p h s p r e s e n te d i n t h i s c o n te x t show th e r e g io n o f
s te a d y -sta te e x te n d in g a lo n g th e g u id e w ith i n c r e a s i n g tim e f o r th e
-1 7 2 ”
s p e c i f i e d f i e l d com ponent,
(5 .1 )
f o r th e d e riv e d component th e r e i s no
a p p ea ra n c e o f th e s te a d y s t a t e f o r th e sm all v a lu e s o f tim e ch o sen
( o f th e o r d e r o f a c a r r i e r c y c l e ) .
The a n a l y s i s h a s b e e n c o n fin e d to a p a r t i c u l a r p a i r o f t r a n s v e r s e
com ponents v /ith i n v e r s io n o f i n t e g r a l s h a v in g k e r n e l f u n c t i o n s , ^ ^ ( s )
an d # 2( 3) .
T hese a re o n ly two o f th e f u n c tio n s c o n ta in e d i n e q u a tio n
( 2 , 3 , 2) f o r th e v a r io u s f^ ^ (s),
A f u t u r e p o s s i b i l i t y i s an e v a l u a t i o n
o f a l l th e f u n c tio n s c o n ta in e d i n t h a t e q u a tio n .
T h is would y i e l d a l l
th e components o f th e f i e l d e x c i t e d by th e u n i t s t e p m odulated
c a r r i e r wave.
U sing th e s u b s t i t u t i o n s o f s e c t i o n ( 2 ,1 2 ) , p =: - ( § + 1/ § ) / 2 j
and (p
2
~
+ 1)2 = ~ ( | ~ 1/ § ) / 2 j >(p = s / w ^ ), th e i n te g r a n d i n th e
i n v e r s i o n i n t e g r a l ( 2 , 12, 6) can be m u lt i p li e d by th e r e l e v a n t f a c t o r
-(g +
1/ ^ ) / 2j , to g iv e th e o u tp u t f u n c tio n s f ^ , where n = 1 ,3 an d A,
( th e case f o r n = 2 w ith th e i n v e r s i o n i n t e g r a l ( 2 . 12, 9) h a s b e e n
c o n s id e re d i n d e t a i l * o f c o u r s e ) .
The s u c c e ss o f t h i s f u r t h e r
a n a l y s i s w i l l depend on w h e th e r t h e § dependence can be e x p re s s e d a s
( Ê^
o r (Ç ^ - l / y ^ )"^ a s , f o r exam ple, i n th e i n t e g r a l
( 2 , 12, 9) f o r f g f z / t ) .
I t i s c o n clu d ed t h a t th e s h i f t o f em phasis to th e s tu d y o f a
p a i r o f t r a n s v e r s e com ponents, p r e s e n te d i n C h a p te r 2 f o r f^ < f ^ ,
h a s h i g ii l ig h t e d some i n t e r e s t i n g p o i n t s ,
3o2
The tim e de p e n d e n t stu d y o f " f r u s t r a t e d T o ta l R e f le c t i o n " .
I n C h a p te r 3.^ th e m icrowave an alo g u e o f th e quantum m e c h a n ic a l
" tu n n e l e f f e c t " h a s b een e x te n d e d from th e s te a d y s t a t e c o n s i d e r a ti o n
p u rs u e d by s e v e r a l a u th o r s to th e tim e dependent i n v e s t i g a t i o n by a
P o u r i e r tra n s fo rm m ethod.
I n th e f i r s t i n s t a n c e , a G-aussian a m p litu d e
-1 7 3 m o d u lated p u ls e was c o n s id e re d .
( 5 .2 )
The r e s u l t s o f a s t a t i o n a i y p h ase
a n a l y s i s and th e c o n d itio n s u n d e r w hich th e y h o ld a re sum m arized
b r i e f l y h e re .
T hese p h y s ic a l c o n d itio n s a r i s e from th e a p p ro x im a tio n s
w hich must be made i n o r d e r to e v a lu a te th e i n t e g r a l s d e s c r ib in g t h e
r e f l e c t e d and t r a n s m i t t e d wave p a c k e ts o r p u l s e s .
T hus, th e
r e s t r i c t i o n on th e i n i t i a l a m p litu d e m o d u lated p u ls e i s t h a t i t s
image f u n c t i o n i n th e fre q u e n c y domain s h a l l occupy o n ly a sm a ll
ran g e o f f r e q u e n c ie s a b o u t
th e c a r r i e r fre q u e n c y , f ^ , and f o r
f^^ < f^ < f ^ g , th e m a jo r p a r t o f i t i s c o n ta in e d v /ith in th e l i m i t s ,
^o1
fo 2 '
The r e s t r i c t i o n s on th e p h y s ic a l system i t s e l f a r i s e th ro u g h
th e so c a l l e d t r a n s f e r f u n c t i o n s ,
|b / a |
and |c / a | , th e r e f l e c t i o n
and t r a n s m is s io n c o e f f i c i e n t s o f th e s e c t i o n o f r ‘’cut-off*"* ^td-db. p llio e d
b etw een two i d e n t i c a l s e c t i o n s o f p ro p a g a tin g g u id e ,
i n te r p o s e d g u id e , f o r w hich d a
f o r a v e iy s h o r t
1 and [c/Al i s a p p ro x im a te ly u n i t y ,
th e tr a n s m is s io n i n t e r a c t i o n tim e , t ^ , i s g iv e n i n e q u a tio n (3 # 6 ,4 ) #
f o r a v e ry lo n g g u id e f o r w hich a a S> 1 and |B /a1
i s a p p ro x im a te ly
u n i ty , th e r e f l e c t i o n i n t e r a c t i o n tim e , t ^ , i s g iv e n by e q u a tio n
( 3 . 6 . 6 .).
I t w i l l be n o te d above t h a t | c/A |
u n ity and in d e p e n d e n t o f fre q u e n c y .
and|B /A | a re a p p ro x im a te ly
T h u s, i t i s e x p e c te d t h a t a lth o u g h
th e tim e d e la y s a re c i t e d above, th e form o f th e p u ls e w i l l be
u n a f f e c te d by i t s i n t e r a c t i o n %?ith th e c u t - o f f s e c t i o n .
I t is
p o s t u l a t e d t h a t f o r a s i m i l a r a m p litu d e m o d u lated c a r r i e r p u ls e and
p h y s ic a l c o n d it i o n s , th e s e r e s u l t s w ould a ls o a p p ly .
f o r th e v e ry s h o r t g u id e , p r e d i c t e d tr a n s m is s io n i n t e r a c t i o n
tim e s w ere fo u n d to be o u t s i d e th e r e s o l u t i o n o f th e m easu rin g sy stem .
T h is was n o t so f o r th e r e f l e c t i o n i n t e r a c t i o n tim e s f o r v e iy lo n g
-1 7 4 -
(5 .2 )
g u id e s and so a m a jo r p a r t o f th e e x p e rim e n ta l e f f o r t was
c o n c e n tr a te d on m easurem ents o f th e s e tim e s .
E x p e rim e n ts were
p e rfo rm ed u s in g s h o r t e r le n g t h s o f in te r p o s e d g u id e f o r tr a n s m is s io n
and r e f l e c t i o n i n t e r a c t i o n s b u t a s th e s e in v o lv e d d i s t o r t i o n o f th e
em ergent p u ls e , th e y w ere a s s e s s e d q u a l i t a t i v e l y o n ly .
The method o f p ro d u c in g th e n an o seco n d c a r r i e r p u l s e s u s in g a
g a te d t , w , t , was ch o sen o r i g i n a l l y f o r p ro d u c tio n o f s u f f i c i e n t l y
p o w e rfu l p u ls e s to p a s s th ro u g h a t t e n u a t i n g s e c tio n s o f c u t - o f f g u id e .
However, d u rin g th e c o u rs e o f th e e x p e rim e n ts and th e t h e o r e t i c a l
i n v e s t i g a t i o n s , i t became a p p a re n t t h a t th e r e w ould b e a s h i f t o f
em phasis from t r a n s m i s s io n i n t e r a c t i o n s to r e f l e c t i o n i n t e r a c t i o n s ,
w ith th e a m p litu d e o f th e r e f l e c t i o n c o e f f i c i e n t b e in g a p p ro x im a te ly
u n ity .
So, w ith o u t th e a n t i c i p a t e d a t t e n u a t i o n , a m ethod o f
m o d u la tio n p ro d u c in g l e s s p o w e rfu l p u l s e s c o u ld be u s e d .
A lso , th e d e te c t i n g and m ea su rin g system i t s e l f a llo w e d d i s p l a y
o f th e r e c t i f i e d e n v e lo p e o n ly o f th e p u ls e on a sam p lin g o s c i l l o s c o p e ,
w ith su b seq u e n t XÏ r e c o r d in g .
I t was n e c e s s a iy to assume t h a t t h i s
system d id n o t a f f e c t th e p u l s e ,
I to ^ ^ ^ ) h a s s u c c e s s f u l l y u se d a d io d e m o d u la to r w hich i n c o r p o r a te d
a c z y s t a l d iode m ounted a c r o s s th e w aveguide, to p ro d u c e c a r r i e r p u ls e s
o f a b o u t ,9 n s e c , h a l f a m p litu d e w id th and v a r i a b le c a r r i e r fre q u e n c y .
U sin g a synchronous d e t e c t i o n scheme i n w hich th e p u ls e s ig n a l i s
m ixed w ith a, v e ry much l a r g e r u nm o d u lated s ig n a l from th e same s o u r c e ,
he was a b le to show t h a t th e s e p u l s e s h a d an
e n v elo p e w hich was
a p p ro x im a te ly G-aussiim i n shape w ith no phase m o d u la tio n .
T hus, n o t
o n ly c o u ld th e shape o f th e en v elo p e o f th e p u ls e be m a in ta in e d , b u t
t h e r e was com plete in f o r m a tio n a b o u t th e p u ls e .
-1 7 5 “
( 5 .2 )
W ith t h i s G -aussian a m p litu d e m o d u la ted p u ls e , a c l o s e r
a p p ro a ch to th e quantum m ec h an ica l tu n n e lin g a n alo g y o f C h a p te r 3
c o u ld be achieveda
W ith com plete in f o r m a tio n o f th e i n p u t p u ls e
( t h a t i s , th e i n i t i a l c o n d itio n i n th e tim e domain) and th e r e f l e c t i o n
and tr a n s m is s io n c o e f f i c i e n t s o f th e sy ste m , th e i n te g r a n d s o f th e
i n t e g r a l s d e s c r ib in g th e o u tp u t p u l s e s c o u ld be known c o m p le te ly .
However, h e re a g a in th e same d i f f i c u l t y a r i s e s a s i n C h a p te r 3 s in c e
th e s e i n t e g r a l s c a n n o t be s o lv e d a n a l y t i c a l l y .
H ow ever, w h ereas i n
th e e x p e rim e n ta l i n v e s t i g a t i o n d e s c r ib e d i n C h a p te r
th e r e c t i f i e d
e n v elo p e o f th e p u ls e o n ly was known, com plete in f o r m a tio n w ould make
th e n u m e ric a l e v a l u a t i o n o f th e s e i n t e g r a l s a p o s s i b i l i t y .
B e a rin g i n m ind th e ran g e o f c a r r i e r f r e q u e n c ie s o f p u ls e s
a v a i l a b l e from th e t . w . t . , t e s t s e c t i o n s were d e s ig n e d w hich gave
p r e d i c t e d r e f l e c t i o n i n t e r a c t i o n tim e s w ith in th e r e s o l u t i o n o f th e
m easu rin g system .
T hese t e s t s e c t i o n s were o f two fo rm s;
firs tly ,
a lo n g s e c t i o n o f u n f i l l e d X band g u id e , i n te r p o s e d betw een tv/o
s e c t i o n s o f d i e l e c t r i c f i l l e d X band w aveguide ;
s e c o n d ly , a lo n g
s e c t i o n o f g u id e o f re d u c e d b ro a d d im en sio n i n t e r p o s e d betv/een two
s e c t i o n s o f u n m o d ifie d g u id e (WG-1i|-).
I t v/as d e c id e d t h a t th e tim e p o s i t i o n o f th e p e a k was th e
m ost c o n v e n ie n t r e f e r e n c e tim e f o r th e p u l s e .
An i n d i c a t i o n o f th e
e r r o r in v o lv e d i n th e m ea su rin g p r o c e s s can be o b ta in e d from th e
r e s u l t s o f tim e o f t r a v e l m easu rem en ts.
The i n c i d e n t p u ls e was
a llo w e d to t r a v e l o v e r a n e x t r a 18cms. p a th le n g th an d th e d i f f e r e n c e
i n th e tim e p o s i t i o n s o f t h e two r e c o r d e d p u ls e s was m easured.
T h is
d i f f e r e n c e was com pared w ith th e tim e p r e d i c te d from a knovfledge o f
th e c a r r i e r fre q u e n c y and t h e group v e l o c i t y i n th e g u id e .
-1 ? 6 ~
(5 .2 )
T here i s a s ta n d a r d d e v ia ti o n o f one i n th e second d ecim al p la c e i n
tim e s w hich a re o f th e o r d e r o f a nanosecond»
T h is i s a ls o th e
d e v ia ti o n o f th e m easured group t r a v e l tim e s from th e p r e d i c te d tim e s ,
A s u i t a b l e tim e z e ro f o r m easurem ent o f th e d e la y o f th e p eak o f
th e r e f l e c t e d p u ls e on i n t e r a c t i o n w ith th e t e s t s e c t i o n o r " c u t - o f f "
g u id e was fo u n d i n p l a c in g a s h o r t c i r c u i t a t th e f r o n t s u r f a c e .
The
tim e d i f f e r e n c e b e tw ee n th e p e ak s o f p u ls e s r e f l e c t e d from th e s h o r t
c i r c u i t and th e c u t - o f f g u id e h a s b e e n d e s ig n a te d th e r e f l e c t i o n
i n t e r a c t i o n d e la y tim e .
The v a lu e o f th e c a r r i e r fre q u e n c y o f th e i n c i d e n t p u l s e , f ^ ,
h as b e e n u se d to p r e d i c t i n t e r a c t i o n tim e s f o r p u l s e s r e f l e c t e d from
th e v a r io u s " c u t - o f f " s e c t i o n s .
T hese p r e d i e te d tim e s can be u se d
to d e s c r ib e th e tim e d e la y o f th e p u l s e , a s a w hole, o n ly i f th e
r e f l e c t e d p u ls e i s u n d i s t o r t e d by i n t e r a c t i o n w ith th e c u t - o f f
s e c tio n .
T here i s agreem ent b etw een m easured and p r e d i c t e d i n t e r a c t i o n
tim e s f o r th e r e l a t i v e l y u n d i s t o r t e d r e f l e c t e d p u lse s*
F o r th e more
d i s t o r t e d p u l s e s , f u r t h e r tim e o f t r a v e l m easurem ents i n u n m o d ified
g u id e were p e rfo rm e d .
These showed t h a t th e c a r r i e r fre q u e n c y was
no l o n g e r t h a t o f th e i n c i d e n t p u l s e , f ^ , b u t h a d a p p a r e n tly b een
s h i f t e d to a lo w e r v a lu e , f ^ ^ , a s d e m o n stra te d by an i n c r e a s e i n th e
group t r a v e l tim e .
I n c o n c lu s io n , i t / . s s a id t h a t th e r e s u l t s o f th e t h e o r e t i c a l
i n v e s t i g a t i o n p r e s e n t e d i n C h a p te r 3 have b e en c o r r e l a t e d s u c c e s s f u ll y
w ith th e r e s u l t s o f th e e x p e rim e n ta l i n v e s t i g a t i o n p r e s e n te d i n
C h a p te r 4«
T here i s agreem ent b e tw een t h e o r e t i c a l l y p r e d i c te d
i n t e r a c t i o n tim e s and e x p e r im e n ta lly m easured d e la y t im e s , w here a
co m p ariso n b etw een them i s f e a s i b l e .
T hus, th e i n v e s t i g a t i o n h a s
-1 7 7 -
( 5 .2 )
gone some way to a n sw e rin g th e q u e s tio n w hich was p o se d i n
S e c tio n (1 # 2 ), nam ely, "How lo n g does i t talce th e e le c tr o m a g n e tic
wave to t r a v e r s e th e c u t - o f f s e c tio n ? "
(A.l)
“178—
( a f t e r C e rrillo * ^ ’'^ ^ )
System o f t r a n s v e r s e m a g n e tic and t r a n s v e r s e e l e c t r i c
d i r e c t v/3>ves i n th e complex fre q u e n c y , s , p lan e
T.H.0 Waves w ith "boundary c o n d itio n
I n i t i a l c o n d itio n
Xu
-1
a+ _ \
llj
àx^ I
\
yA
E ,)
=
Q
( o , s ) (s
( 0 ,s )
/ 2
± -z (s
+ w ^ )2 .e
o
2\ 2/; 7C
p
/k ^ c
E_ z=
/ 2
2 s2/
- z ( s + 0) ; / o
( O ,s ) .e
^
H.
-1
ÏÏg
âXg
V^es A , (o , s ) . e “ z ( s ^ +
Hg =
h^
0
p
/k ^
ôx^
- 1
f ie
I n i t i a l C o n d itio n
1
E-
(A . 2 ) .
•179-
T.H . Tfaves
S’)
^
- ^-z A l ( 0 , s )
I
ô"x.j I I
/ 2
2 \ 2/
- z ( s + 0) ^ ) / c
/kg
'A^(0,s).e
1
_
a*.
E
0
o
- z ( s ^ + »^)?'c
= ■*■
^ 3 (O js )
L.
2
h
«1 =
o.e
/(s
o i
+
#3
ôx.
- z ( s ^ + 0 )^ )/c
e
(s
1
Hg =
11
.
1
=
0
+
0)
C
ôx
c
=
)
(A.3)
•180-
ToEo Waves
B oundary o o n d itio n ;
= 0
I n i t i a l o o n d it i o n
H-z)
^
=
G% B? (O, s )
r
1
" i “ ïÇ
V
•z(s^+ oo^)^c
E3 " ^
1.
k
c
B (0 _ js).e
^
ôx^
u
E-,
0
=
e
^ = Vl-i-e
fi
=
Ô3C>|
(3% + W f)"
.1
C
1/
•z (s^ +
m
=-83 B^ ( 0 , 3 ). e
z ( s ^ + (0^)%
3^( 0, s),G
'181
(A .ij.)
TaS, Waves
Boundary c o n d itio n
V
I n i t i a l o o n d itio n
'
H i)
3
n
z=0
E
= 0 , Bl (O ,s )
3 3
/ 2
2 \ p,
- z ( s + 0) j / c
(0, sXe
°
(j. Cs
V
=
H
1
ÔX
-z (s^ + ^ l ) f c
^B^ (0^s).e
k'
H,
J
2
=
63
b;
( 0,
8)
c. e
( /
+ 0,2 )-
2 1
( A .5 )
-1 8 2 -
C e r r x llo ^ ”'^^ fo u n d t h a t
i f a l l th e i n i t i a l c o n d itio n s i n th e
s domain w ere a r b i t r a r y and in d e p e n d e n t o f each o t h e r , th e s tra n s fo rm e d Maxwell* s e q u a tio n s w ere n o t s a t i s f i e d .
Thus i t was
n e o e s s a ry to f i n d th e p r o p e r r e l a t i o n s h i p f o r th e i n i t i a l c o n d it i o n s .
Por
c o n v e n ie n c e , th e i n i t i a l c o n d it i o n s were d iv id e d i n t o two ty p e s :
F i r s t l y , i f A ^ (o ,s) and
( o ,s ) a re g iv e n a s th e in d e p e n d e n t
i n i t i a l c o n d itio n and B ^ ( o ,s ) = B ^ ^ ^ ^ (o ,s ) = o , th e f i e l d i s
e le c tr o m a g n e tic i f
\
H ^ (o ,s )
=
- s e A ,( o ,s ) y
1
'
K
6^2
a
( A . 5.1 )
0X^
K
E ( o ,s )
=
^2
( o ,s ) /k 2
§1^2
T hese can be s u b d iv id e d f u r t h e r f o r
A ^ (o ,s)
o,
A _ (o ,s ) = o ,
^
^
A'
( o ,s )
5 ( 2)
S e c o n d ly , i f B ^ ( o ,s ) and
^
or
o
B ^ ^ ^ ^ (o ,s ) a re g iv e n a s th e in d e p e n d e n t
i n i t i a l c o n d itio n , A ^ (o ,s ) = A ^ ^ ^ ^ (o ,s) = 0 th e ü e l d s a re
e le c tr o m a g n e tic i f
(A . 5)
-1 8 3 -
E .( o ,s )
h.
0^3
0 x,
S}iB ( o , s ) / k
E „ ( o ,s )
J
( A .5.2)
i
0G3
0X,
/
JL
hg
H o (o ,s )
^^3
2;
A gain^ t h i s can b e s u b - d iv id e d i n t o
B ^ ( o ,s ) i
o,
® 3 (z)^°^ ^ ) = o
B^Cojs) ~
Of
B^^^^(oj,s) = o ,
and.
--I81).A ppendix B.
(B .1 )
a f te r C a rrillo «
R e s t r i c t i o n s on th e a n a l y t i c a l form o f th e s - tr a n s f o r m e d
i n i t i a l c o n d itio n s
----------------------- — ^
A’ /
\and B*
3 ( o ,s )
To e n su re t h a t th e s o l u t io n s
and
/
3 lo ,s ;
be e le c tr o m a g n e tic ,
G e r r illo ^ ^ ^ ) h a s fo u n d i t n e c e s s a r y to im pose r e s t r i c t i o n s on them
a t t = z /o .
The i n i t i a l c o n d it i o n ,
( o ,s ) o r A^ ( o ,s ) i s
c o n s id e re d i n some d e t a i l and t h e com ponents d e riv e d from t h i s a re
g iv e n i n A ppendix A (A. 1 ) •
The f i r s t s te p
i n th e d e r i v a t i o n
o f th e s e
r e s t r i c t i o n s in v o lv e s
p ro o fs o f th e fo llo w in g two th eo rem s f o r th e f u n c t i o n , f ( t , k ) .
f ( t,k )
=
o
fo r t< k
o
fo r t> k
If
..(B * 1 * 1 )
and th e f u n c t i o n and i t s d e r i v a t i v e s w ith r e s p e c t to t and k a re
■^(t)
th e n :
Theorem I
i
vCf 4jco
s P ( s ,k ) ,e x p ( s t ) .d s
d-jco
T h e o re ^ n
f | '3^^)(t,k) = f ( k , k ) l ( t - k ) + ( l / 2 r r j )
\
2 ^ ^ ^ ( s ,k ) e x p ( s t ) . ds
where E( s) i s th e L ap lac e tr a n s f o r m o f f ( t , k ) .
U sing th e s e theo rem s
i t i s p o s s i b l e t o show t h a t th e f i e l d c r e a te d by th e i n i t i a l c o n d itio n ,
E ^ (o ,s ) =
£
A ^ (o ,s ) v d .ll be e le c tr o m a g n e tic i n th e t dom ain i f
^f
^
z /o “ ^
••ooeooo**
=
s /c = °
(B*1 # 2)
-1 8 5 C l(= 1 ,% 2,
(B.2)
^ 2 /0 +
z ,t) ^ = s /c = °
The n e x t problem i s to f i n d w hat m ust be th e a n a l y t i c a l
s t r u c t u r e o f E ^ (o ^ s) ( o r t j ^ ^ ( o ; s ) ) i n o r d e r to f u l f i l ( B ,1 .2 ) ,
It
w i l l be n o te d t h a t th e above r e l a t i o n s h o ld f o r th e i n i t i a l v a lu e s
o f th e com ponents a t th e vfavefront^ t = z/o*
Thus th e n e x t s te p
in v o lv e s th e a d a p t a t i o n o f th e " i n i t i a l v a lu e theorem " i n L ap lac e
tra n s fo rm th e o z y to a su ita b le ? form f o r th e ty p e o f f u n c t i o n
p ro p o se d i n
U sin g theorem I , i t can be shown t h a t f o r th e f u n c t i o n
f ^ ( t ^ k ) w ith tr a n s fo r m E ^ ( s ,k )
=^n
-k (s^ +
l im i t^
(s ( ÿ ^ (s ,s ) ) = f ^ ( t ,k ) ^ _
(The f u n c tio n s 3?^(s,k)« (n = o ,
( 2 ,3 * 2) ) ,
, ,,...,,..( B
.2 .1 )
2, 3» k) a re l i s t e d i n e q u a tio n
I t i s assum ed t h a t f o r l a r g e v a lu e s o f s ,
= M/s^ ,
w here M = c o n s t a n t , y = i n t e g e r .
S in c e
% ,t) =
A ^ ( o ,s ) . e x p ( - z ( s ^
( p^ =
th e n
(x^ Xg) A ^ (o ,s ) = E ^ (o ,s )o
T h e re fo re , i n o rd e r to s a t i s f y
th e f i r s t c o n d itio n i n (B*1 . 2 ) , l i m i t „
( s / i ^ ( o , s ) ) -) o . T h e re fo re i t
'7
S^JCO
J
2
i s r e q u i r e d t h a t A ^ (o ,s ) b eh av e a t l e a s t a s M/s a s 8-)co .
C e r r i l l o f u r t h e r p ro v e s t h a t i f A ^ (o ,s ) ( o r E ^ ( o , s ) ) i s th e o n ly
i n i t i a l c o n d itio n o f e x c i t a t i o n , a l l o t h e r s b e in g z e r o , th e n
^ 2(^ 1,
=
./c
= z/o
=
°
=°
S in c e z can b e any c r o s s - s e c t i o n i n th e g u id e , i t c an be z = o .
A t z = 0 , s in c e A ^ ^ ^ ^ (o ,s) i s z e r o , th e n from e q u a tio n (A .3*1) ;
"186"
(B .3 )
Then l i m i t a
( s E ,( o ,s ) ) = o
s-^co
1 ' '
and l i m i t a
( s E ^ ( o ,s ) ) = o
8-4m
2 ^
.
.
E i ( x i ,% 2,
A3( 2)(°>® ) = °*
G ) t) t
2/ c
_
^
=
z /o
“
°
T h e r e f o r e , H ,(x ^ ^ ^ 2 / ' ^ h = z / o “
'^-2(^1 ,^ 2 , ^ ’ * h = z / o = °
S in c e th e t r a n s v e r s e com ponents m ust v a n is h a t t = z /c ^ t h e i r
tra n s fo rm s m ust ohey c o n d itio n (B*2*1 )»
Use o f th e e x p re s s io n s on
(A o1) y i e l d s th e f o llo w in g c o n d itio n
s
—^ M/ w h e r e
l i m i t g _ ^ ^ (E j ( o , s ) )
Y 55-3p
I t can b e s i m i l a r l y shown t h a t i f H ^ (o ,s ) i s g iv e n , a l l o t h e r s
b e in g z e r o , l i m i t ^
«'
Y ^ 3*
(H^(o
s —>cci
3
.s))= =
"
lim its
( B ^ ( o ,s ) ) - ^ N /
8 -)CD ' 3
T h is i s c o n d itio n I g iv e n i n s e c t i o n ( 2 . 4)0
^
where
C o n d itio n I I
can b e s i m i l a r l y e s t a b l i s h e d .
C o n d itio n I I I s t a t e s t h a t i f A ^ ^ ^ ^ (o ,s) o r B ^ ^ ^ ^ ( o ,s ) i s
g iv e n r e s p e c t i v e l y a s th e
i n i t i a l c o n d itio n , i t i s
e q u iv a le n t to
s p e c if y in g th e t r a n s v e r s e e l e c t r i c o r t r a n s v e r s e m ag n etic component
o f th e f i e l d a t z = o.
The j u s t i f i c a t i o n f o r t h i s i s fo u n d i n th e
l a s t two e q u a tio n s o f (A .3*1) and (A .3 o2)
' *i 8'7*''
H efereiaoeS o
(1 )
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33
1956.
-190"
;om ents,
The a u th o r would l i k e to e x p re s s h e r thanlcs to D r. KoVf.H. Moulds
f o r h i s g u id an c e th ro u g h o u t th e c o u rse o f t h i s w ork;
th e S c ie n c e R e se a rc h C o u n c il and th e U n i v e r s i ty o f S u rre y f o r
f i n a n c i a l s u p p o r t;
th o s e members o f th e t e c h n i c a l and com puter s t a f f a t th e
U n i v e r s i ty o f S u rre y , v/ho have g iv e n t h e i r a s s i s t a n c e a t v a r io u s
s ta g e s o f th e w ork;
M u lla rd L im ite d , f o r d o n a tin g s e v e r a l
LB6/10 t r a v e l l i n g wave t u b e s ;
M rs. H odges, o f M alvern f o r ty p in g th e s c r i p t .
The a u th o r w ould l i k e to th a n k h e r husband f o r h i s
en co u rag em en t, and a d v ic e on many a s p e c ts o f th e w ork.
F i n a l l y , th e a u th o r i s in d e b te d to th o s e who c a r e d f o r h e r
young d a u g h te r.
W ith o u t them , she w ould n o t have b e e n a b le to
u n d e rta k e t h i s w ork.
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