109
RADIATION CHAitiCTElUST1C.S OF SOLID RECTAIUJLAR DIELECTKIC HORNS
ri
Chatterjee
,K
G
Narayanan
,
and
A
Kumar
ABSTRACT
The r a d i a t i o n c h a r a c t e r i s t i c s of s o l i d d i e l e c tri c horns of r e c t a n g u l a r cross- sec ti on,
e x c i t e d by a r e c t a n g u l a r m e t a l waveguide
o p e r a t i n g i n t h e doninrant TE
mode have
been s t u d i e d both t h e o r e t i c a 8 y and’experimentally. The t h e o r e t i c a l a n a l y s i s i s based
on the modal knowledge of t h e f i e l d distributions around t h e d i e l e c t r i c s t r u c t u r e and
over t h e free- end aperture.
The e f f e c t of
t h e feed-end r a d i a t i o n i s supernosed s u i t a b l y
on t h e free- end r a d i a t i o n t o o b t a i n t h e t o t a l
f a r - f i e l d p a t t e r n . Experimental v e r i f i c a t i o n of t h e t h e o r e t i c a l r e s u l t s i s obtained
by probing t h e n e a r - f i e l d a s well as by t h e
measurement of t h e f a r - f i e l d p a t t e r n a t h
13.2 cm and h = 3.2 nun.
1.
INTil(lllUCT1ON
D i e l e c t r i c r o d s of r e c t a n g u l a r c r o s s - s e c t i o n
have been r e c e i v i n g g r e a t e r a t t e n t i o n in
r e c e n t times with t h e advent of p l a n a r m i l l i meter wave i n t e g r a t e d c i r c u i t s and of integrated optics.
Several a p p l i c a t i o n s as r a d i ators and waveguide components have been
aeveloped. L i g h t weight d i e l e c t r i c antennas
of r e c t a n g u l a r cross- section have been studi e d by Kobayashi e t a 1 ( I ) , Sen and Chatterj e e ( 2 1 , P a u l i t ana C h a t t e r j e e ( 3 ) and
o t h e r s . d a d i a t i o n from pyramidal d i e l e c t r i c
waveguides has been s t u d i e d by Brooking e t
a 1 ( 4 ) . I n t h i s paper i s presented t h e
r e s u l t s of an a n a l y t i c a l and experimental
study of t h e guidance and r a d i a t i o n c h a r a c t e r i s t i c s of s o l i d d i l e c t r i c r e c t a n g u l a r
horns c a r r y i n g t h e ifl mode.
2.
CUIULD MODcS U‘4 T ~ LtUCTkNGULHti SOLID
UIELtCTKIC HOiW
The boundary-value problem of t h e r e c t a n g u l a r
s o l i d d i e l e c t r i c horn \ F i & . 1) has been s t u d i e a by modelling it a s a cascade of piecewise uniform r e c t a n g u l a r d i e l e c t r i c guides
(Fig. 2). Using f i a r c d t e l i ’ s ( 4 ) method, t h e
following c h a r a c t e r i s t i c e q u a t i o n s have been
modes:
derived f o r dy
P9
where kx, kx2 and ky, ky2 are t h e propagation
constants i n t h e x a d y directlons i n s i d e
t h e u i e l r c t r i c rod and i n r e g i o n s 12,4) and
(3.5).
b m i l a r l y t h e c h a r a c t e r i s t i c equations f o r
modes are
For hybrid modes, eqs. ( l ) , (2). ( 3 ) and (4)
a l l hold good. The t r a n s v e r s e and 1 o n Q t u d i n a l propagation c o n s t a n t s have been o b t a i n e d
by s o l v i n g t h e c h a r a c t e r i s t i c equations
g r a p h i c a l l y and numerically f o r horns of
s m a l l f l a r e angles upto 60. Fig. ( 3 ) shows
t h e t h e o r e t i c a l and experimental v a r i a t i o n
of E o v e r t h e free-end a p e r t u r e of a pyram i d d horn antenna and Fig. ( 4 ) t h e theoret i c a l and experimental v a r i a t i o n of k along
t h e l e n g t h of t h e horn f o r another p 3 a m i d a l
horn antenna f o r t h e EY, mode. I t can be
seen t h a t t h e agreemen4 between theory and
experiment i s good.
C a l c u l a t i o n of t h e average power flowing
down t h e horn i n s i d e and o u t s i d e , shows t h a t
l e s s than 5% of t h e t o t a l power flows outs i d e t h e horn, when t h e l e n g t h of t h e horn
exceeds about 5
Fig. 5 shows t h e c a l c u l a t e d power confinement Versus l e n g t h f o r
an X-band horn and a 95 CHz band horn.
.
The measured phase e r r o r over t h e f r e e end
of t h e horn i s l e s s thzn 250 f o r small angle
horns; hence t h e assumption of a plane wave
propagating along t h e a x i a l d i r e c t i o n i n t h e
horn i s s a t i s f a c t o r y . For a p e r t u r e dimensions g r e a t e r than 3
t h e phase e r r o r i n c r e
a s e s and hence t h e propagating wave becomes
more s p h e r i c a l .
,
3.
THEOitETICkL iiADIATIGN CHAhCTERISTICj
The two- aperture theory used by &-own and
Spector (5) and by James ( 6 ) has been used
t o d e r i v e t h e r a d i a t i o n c h a r a c t e r i s t i c s of
t h e r e c t a n g u l a r s o l i d d i e l e c t r i c horn.
The
horn i s assumed t o be an e f f i c i e n t s u r f a c e
waveguide t o t r a n s p o r t a l l t h e energy launched i n t o i t i n t h e form of guided modes
u p t o t h e f r e e end from where i t i s r a d i a t e d .
Some energy which f a i l s t o be trapaed by t h e
t o t a l i n t e r n a l r e f l e c t i o n phenomenon i n t h e
guide i s r a d i a t e d o u t of t h e d i e l e c t r i c
s t r u c b x e c l o s e t o t h e f e e d end and i n t e r f e r e s with t h e free- end r a d i a t i o n a t t h e
far field.
Z,1
2 a d i a t i o n due t o t h e Free-end Aperture
The r a d i a t i o n f i e l d due t o t h e f i e l d s d i s t r i b u t e d t a n g e n t i a l l y over t h e free- end
a p e r t u r e plane of t h e d i e l e c t r i c horn i s
ootained by t h e s c a l a r d i f f r a c t i o n i n t e g r a l .
The d i f f r a c t e d f i e l d a t any p o i n t P due t o
d d i s t r i b u t i o n of f i e l d s on an a p e r t u r e i s
given by S i l v e r (7).
up= %IT
& F l u , 3) en p I-jk,$/!fik +) iz . r~
-
4
+
where F ( X , Y ) =
( x , y ) exp [;j\vlz>a,3
A( x, y) being t h e a a p l i t u d e d i s t r i b u t i o n and’
-4-tx.Y) t k e phase d i s t r i b u t i o n over t h e a p t u r e A, n i s t h e u n i t v e c t o r in t h e direct i o n of t h e f i e l d p o i n t P from t h e p o i n t
( x , y J on t h e a p e r t u r e p l a n e , and d i s t h e
u n i t v e c t o r n o r m d t o t h e wave f r o n t a t t h e
a p e r t u r e p l a n e towards P.
-
The phase (x,y) a t t h e p o i n t (x,y,zt) on
t h e free- end a p e r t u r e plane z
zt i s given
bY
being t h e r a d i a l d i s t a n c e f r o m t h e Origin
t h e p o i n t (x,y.z). i f t h e d i s t r i b u t i o n of
phase over the a p e r t u r e i s assumed t o be
t h a t corresponding t o t h e Presence of a sphe
rical wave a t this plane, i n a manner simil a r to t h e treatment of a small m e t a l l i c
horn as given by C o l l i n and Zucker (8). Eq.
(6) i s used i n e v a l u a t i n g eq. (5).
H
t8
3.2
Radiatim due t o t h e Feed-end
The r a d i a t i o n from t h e feed-end is assumed
t o be t h a t of t h e open-ended rectangular
metal wave-guide launcher, as modified by
t h e presence of t h e dielectric i n f r o n t of
it. A r e l a t i v e amplitude weighting f a c t o r
which g i v e s t h e b e s t f i t with t h e experiment a l l y observed p a t t e r n i s determined by a
t r i a l and e r r o r method, and this is used
while superposing t h e f eed-end r a d i a t i o n and
t h e free- end r a d i a t i o n . If the power flow
i n t h e launcher metal waveguide carrying the
Th10 mode is P
t h e power flow i n t h e rectangular d i e l e c g i c guide propagating i n t h e
67 mode i s PD, and i f t h e power t h a t can be
radiated o u t of t h e free- end plane i s PT,
then
whereT- power r e f l e c t i o n c o e f f i c i e n t a t the
feed-end and m i 8 t h e f r a c t i o n of the t o t a l
power PL t h a t escapes t h e dielectric guide
by radiatim a t t h e feed-end junction. The
maximum value PTmX of PT i s given by
4.
EXPFRIWNTAL
The observed r a d i a t i o n p a t t e r n s of rectangular
s o l i d d i e l e c t r i c horns e x c i t e d by a simple
open ended metal waveguide launcher have a
c h a r a c t e r i s t i c side- lo be s t r u c t u r e with
spaced m i n i m a r e s u l t i n g from t h e i n t e r f e r e n c e
e f f e c t s of the feed-end r a d i a t i o n and t h e
free- end radiation.
The p a t t e r n s a r e f a i r l y s i m i l a r i n t h e E and
H planes. The maximum gain is about 6 db. The
cross-po_larized f i e l d components are q u i t e
small. &he p o s i t i o n s of t h e f i r s t few minima
and t h e gain along t h e boresight d i r e c t i o n
are s e n s i t i v e f u n c t i o n s of t h e f l a r e angle of
t h e horn, for a given horn length. These
c h a r a c t e r i s t i c s suggest t h e p o s s i b i l i t y of
u t i l i z i n g such antennas i n s p e c i a l i z e d applications.
Fig. 6 and 7 show experimental and t h e o r e t i c a l
r a d i a t i o n p a t t e r n s of t w o horns a t A I 3.2 c m
and
I 3.2
rnm respectively.
5.
CONCLUSIONS
The agreement between theory and experiment
i s good. The assumption of t h e EY1 mode a t
t h e terminal plane gives s a t i s f a c t o r y r e s u l t s
angle
f o r t h e r a d i a t i o n patternsof small
horns, while it i s necessary t o consider
higher order modes f o r l a r g e a p e r t u r e horns.
UFEFWCES
1.
Kobayashi S. e t a l , 1982, ' D i e l e c t r i c
tapered r& antennas f o r millimeter-wave
a p p l i c a t i m s ' , IEEE Trans.knt. and Prop.,
V O ~ . AP-X, Noel, pp. 52-58.
2.
Sen, T.K. and Chatterjee, R., 1978,
'Rectangular dielectric rod T t microwave
frequencies
P a r t s I and I1
Inst.Sci.
Vol. 60, No.5,pp. 9i%&
!:!
3.
P a u l i t , S.K. and C h a t t e r j e e , R . , 1983,
'Radiation c h a r a c t e r i s t i c s of tapered
rectanrmlar d i e l e c t r i c r o d antennas a t
4.
Brooking. N. e t a l , 1974. 'Radiation
Datterns of D W d d a l dielectric waveguides', E l e ' c b o n i c s L e t t . , Vol. 10,
NO. 3 9 ~ ~35-34.
.
5.
Brom,J. and Spector, J.O.,
1957, 'The
r a d i a t i n p r o p e r t i e s of end- fire aerials;
Proc.
Vol. 104 B, pp. 27-34.
6.
James. J.R., 1967, 'Theoretical i n v e s t i g a t i o n s of c y l i n d r i c a l d i e l e c t r i c antennas'. Proc. I=, Vol. 114, No.3,
PP. 309-319.
7.
S i l v e r S., 1949. 'Microwave Antenna
Theory'and Design', Radiation Laboratory
Series
158-162, and
8.
C o l l i n , H.E. and Zucker: J. 1966
'Antenna Theory, P a r t If
M&raw'Hi&
Chap. 21.
Narayanan,K.C.,
1984, 'Radiation charact e r i s t i c s of rectangular dielectric horn
antennas',Ph.D.Thesis,Ind.Inst.Sci.,
Bangalore, India.
-
(9)
Using e q i ( 9 ) ) t h e maximum r a t i o of t h e f i e l d ampllnide c o e f f i c i e n t C of t h e ET1 mode
to t h e f i e l d amplitude c o e j f i c i e n t A,0 of
t h e 3%
mode can be evaluated. T h i s r a t i o
tC1/AIA? has t o be reduced to o b t a i n t h e
best match of t h e t h e o r e t i c a l r a d i a t i o n patt e r n with t h e experimental pattern, when the
feed-end and free- end r a d i a t i o n p a t t e r n s are
superimposed t o o b t a i n the t o t a l r a d i a t i o n
pattern.
Uue t o r e f r a c t i o n e f f e c t s a t t h e boundary
s u r f a c e s of t h e d i e l e c t r i c horn, t h e r a d i a t i o n from t h e feed-end i s modified i n two
ways as follows: ( a ) The d i r e c t i o n of t h e
emergent ray is modified and this shifts t h e
angular p a t t e r n of t h e f e e d a p e r t u r e (b) The
amplitude of t h e f i e l d s emerging o u t of t h e
d i e l e c t r i c i s modified d i f f e r e n t l y i n differ e n t d i r e c t i o n s , according t o the Fresnel
r e f r a c t i o n c o e f f i c i e n t . Quantitatively, t h i s
r e s u l t s i n t h e n e c e s s i t y of multiplying t h e
O r i g i n a l feed-end r a d i a t i o n p a t t e r n in t h e
4 ='0 and 4 I goo p l a n e s by the f a c t o r s T
and T r e s p e c t i v e l y as given by Narayanan(8).
P
Id,
9.
;
a=a,+ 22 tan e,
A=&+Zzptan e,
b -b,,+2z
tan eg
//-F\
I
B = b,,+2zptan OB
or,%- se mi-f tare
angles of horn
w"
-12
v
0
0
a
-12
1
I
2L
I
I
16
.
I
8
1-la
0
-x.mm-
,
8
,
,
16
,
,
2L
(b)Pyramidal horn eA-2.5', eB-2.50,L -35cm
E, i 2 . 5 6 ,
1 = 3 . 2 cm.
- Theory Eyl
FIG.l RECTANGULAR DI
CTRlC HORN
ao= 2 . 2 8 cm bo=l.Ocm
I
Measured
FIG.3.VARIATION OF Ey OVER F R E E END APERTURE
GEOMEtEY
FIG. 2a. APPROXIMATION O F DIELECTRIC
HORN BY CASCADED UNIFORM GUIDES
c
8
-
Horn-$=2.56,oA=1.5'
g = l . S * . X =3.2cm
Field E y
Location- 1.0
y x b h + O3mm
-
o-Theoretical, E i Mode
0 - Measured phase
shift
(b)
(dlmmersed in surrounding medium
(b)Regions of field distribution
FIG. 2b. RECTANGULAR DIELECTRIC
G U l E IN CROSS SECTION
-*0
420
280
E
E
- 100
-
N
- 60
1
b
y$l
70
k, RAD
FIG.4.VARIATION OF FIELDS 8 k r ALONG
THE LENGTH OF THE HORN(PYRAMI0AL)
g 201
%
, 2i5b
5
p o
I
15
20
t.cm
10
QG
, pb.,
1
25
M
3s
~.
(a) x-Band dlctectric horn
0"
.c
(P)-Perspex.€,=2.56
(S)-Silicon. E, ~ 1 2 . 0
A =3.2rnm
601
--
12--14
161
0
z.mrn
(b)95 GHZ-Band dielectric horn
FIG.5. POWER FLOW IN RECTANGULAR
DIELECTRIC HORNS
' t
401
0
24
12
1
1
I
36
L@
60
Angle. degrees
€,a2.56
@,s2.5*
1 =3.2crn. 0,=2.5'
L =30.Ocm. I=O'Plane
Feed-end refraction included
I
FIG.6.RAOlATlON PATTERN OF
RECTANGULAR 01EL E CT RlC
HORNS
Theory .feed-end refraction included
Erperiment
1
I
I
12
I
24
48
36
60
Angle, degrees
E Sectoral horn, silicon (100 ohm-cm)
F I G . 7 . RADIATION P A T T E R N S O F
DIELECTRIC H O R N S AT
95 GHZ B A N O .