Journal of the Indian Institute of Science, Bangalore 560 012, India
[Section A : Engineering and Technotogy]
Jrmrulry !980
CONTENTS
ORIGINAL PAPERS
Microwave Engineering
Pri~e
COSINU~OIDAI,LY
MODULATELI SURFA.CE WAVE STRU<:T~~ES-NIIMERICAL CAI.C[~LAS. R~rkmiziutzd S . K. C/rat/ccj~~r
TlONS OF CHARACTCRISTICS
x.
SHORT COMMUNICATION
Chcrnica' Engineering
AM BQ~JA'PION OF STATE
BOOK REVIEWS
FOR SOME G A S U
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pl.~cts, 1::. ''ihrltm &,d;
i d m e , w r ~ d , , , " , A h.1 v! t i t ? w:~utc:~:s A , : l ~ :kw t y ( n ~
in h h l e qucing on a rip.mte rbcut in thi urdci :,I tiair i.ict.rrts e t r i rli; cr?! 2 z . i .ii,i.t: v.1
~r rhc end of rhc piper. 'The tclciemx: d i w l J hr, j i r t r , r i ititit :,;I1 ic:irir
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cr:impio :
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132. ' * I - la!!, !':rwi..c,
Hnli, New Jersey.
(9) C;nngJhrrriah, T., Tbo <;barxdmit:ir of Seilr?u.dr,J I:ir:i., l!I:.l, ( fr .". pii.
25-27 Thesir (unpublisheJ), Indian Invritt~cc ut bfivarr.
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~
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Standard mathematicnl notations s h u u l bc adhemi m. Plcrre ,ii>trr:;rl;irli bcr:vrcn %A)" and
"kappa", "ell" and "one" and other similsr symbolr likely ro ~ i u r c.onfurir;rr.
Abbreviations, such as ez., ai 51 snd LC., t i n he used, if rmii+no~lur,l ~ l r b r ~ u t r r i ~ ~ d
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J. Indian Inst. Sci. 6; (A), Jan. 1980, Pjx 1-36
@I
Printed in 1nd1.i
Cosinusoidally modulated surface wave structures-numerical calcuIations
of cliaaracteristics
T. S. RUKMlNl
AND
S. K. CHA'r'TEKJEE
Departtitcnt of tlcctncal Co~ninun~cation
Enginwring, Indian institute of Sci~ncc,Bangalore 560 012,
India
Abstract
Results of con~putcr-aidedc;lloubtions for Ithc powcr carried by surface waves, atlentation, swface
impedance and tho cikcts of nwdulatiotl index on t t c radial lield dccay, spatial bai-monics, otc., arc
reported. Stability of surfucc wavc modes confu'tncd by cxperinlcnlis disct~ssodwith thc help of stability charts drawn for thc rno.lulatcd structures.
Key words : Molulated surfuco w~vost~cturcs,
stability of surfacc wave rnodos.
1. Introduction
The paper prescnts a report on the nutnerical computations of the characteristics for
cosinusoidally spatially modulated circular cylindrical metallic corrugated structures
excited in E, mode based on the theoretical analysis by Chatterjee et u P 3 . It is shown
that spatially modulated surface wave strwturcs with low values of modolation index
(6 <: I) can support more strongly bowd surface waves than uniformly corrugated
surfase wave structures.
2. Equivalent dielectric constant
E~
The corrugated region for an uniformly corrugated rod excited in g wave is simalated,
as a homogeneous dieleztrio region of cqnivalent dielectric constant eo =f (s, b, a)
where s, b and a denote respectively the spacing between any two adjacent discs, radius
of d i m and radius of the central supporting rod. Since it has been proveds that the
relative amplitndes (A,) of Floquer harmoniorsatisfy Lhe inequality relation A, ( m f a)/
Ao (m = 0 ) < 1, therefore, ea for the fmdamentaI mode (m = 0) is calculated by using
the relation
where, tilt. p h a ~ ec o i ~ \ u ~ !I,,
n t fur the Sitndante~ii;ilmudc i' itctcr~l>inodl'rcriil rhc rolution
of tlw following equi~tion'
Figure I ~ h o w sthat
(iii) For a groove deplh % ?d,,/?,
G" ilpproachch the v:tluc crl' i r n t r ? which i s the villuc
(air). Thiz implies (hat 11 rnodiliad ' Harms-Gowhau '
(hr the externnl n~cdiu~n
line approaches ' Sommerl'eltl ' surf;ue w;we line when the gruovc depth for r
spatially modulated line app~owhestho valuo ol' inkryriil rrtutriple (rr - 1,2,3.. .)
of half wavelength (1.,,/2).
(iv) It is evident that increase of 2:" i s associatccf witlr inclxasc <11' ji,,. l'l;ereli~re. .2
slr~rciurewith highcr values of sat1 supporl nlorc strongly bound w r f a x wave
which is in conformity with the case of Harms-Goubat~ line.
(v) co in-rwses w;Lh k,, i.c., with increasing frcq~icncy oS crcitatiun Ibr ; ~ l l o d
values o f b and s. Hence, for it surhce wave structure w i t h a n~odilirds u r f m
more strongly hot~ndsurrace wave can be suivported with inae:iaing frequency.
This is also in conformity with the conventional W-6. line.
Hence, it may be concluded that the evolvction of tho equivalent Jiciwtric cOnLept f ~ r
the corrugated region for the purpose of anrtlysis is justified.
3. Modulation of cu
Since go +f(& b) with the parameter a held constant, spatial or depth moduk~tionis
achieved by varying s or b respectively and keeping tho other parameter constant. 111
lhis case, spatial cosinuroidal modulation is studied.
FIG.1 (40. r " w b with
S as panimetcr.
where the modulation index d
ir
delincd by
4. Radial propagation constants
Using the €allowing mixed boundary condition%
Fro. 2 u. Spacine vs
9 = 0.2 m.
2dIL and
6,
vo 2m:IL far cam 3.
rb
-
2.914,
b = 0.3 cm, S = 2-05 cm,
MOWCAPEII SURFACE
WAVE STRUCTURES
7
and appropriate field components for the two media the following characteristic equation"~ ohtained
Fro. 3 a. Variation of kt and k, with 6 at z P.0 with
toas
parameter.
The solution of (5) will1
kg
-
kr
+ k3, [I -
t
(z)]
yields thc values of k , and k, asf'(d) for fixed voltres of re (Fig. 3), st :-. 0. Figure4
shows t'le variation of k , and k , at different Iw:~tioncof the .;!ruulure ever a half period.
Figure 5 shows a comparison bctwecn k t , and k,,(p: order ol' mode.;) between two
successive higher order modes.
The following observations may be made regayding k , nnd k , :
> k , for [he slrirctwe with .$ -: 2.0 over I ~ whole
C
r a n g of (5 hut for structures will1 c0 := 1.062, 1.293 and 1.323, k, :?. k, over only n linlited range of 8.
(i) k,
I(z) and hense /
l
are slowly varying functiotln o l ' 2 n ~ ! 1i.c..
~ . the locations
along the cell length 2.
(iii)The different natnre of variatiolt or k , and k, with Zrrzjl: far the two modes
signifies that the two modes behave differently with regard to rndial field decay
at different locations of the stntctups,
(ii) k , and
.
.
,
MODULATED SURFACE WAVE STRIJCTURES
I
Plo. 4s. Variation of kt. k, witlt 8 for cu
-
-
1.323 o w lie period InzlL
t-
3.0
"0
1.0
-
ZTTZ/L
PKI. 4 6. Variatlon of k,, k, with S ror
--
2.0
3.0
- 1,063over the period I n j L .
(iv) The constants k,,, ke, and /)6 i n the two media forp modes arc related as follows:
fl; = k':,, k t (7) First medium (med. 1)
and
3/; =- Gkf,
-k kt c ( z ) (8) Swond (modula~ed) medium (med. 2).
The equalities k,, = k,,, k,, - k,, at LI i=i 2.35 I~/2flsignify that the two modes
beooma indistinguishah10 (8' = P,).
-+
(v) The equality k,, -= k,, signifies that
which means that fit > j, since 6 (2) > I , or in other words. I*,, :.. 11,~. Hence,
the higher order mode is more slrongly hound and consequenrly its rare of
radial decay is faster than that of the first mode.
(vi) The equality kcT = k , , = k,* signifies not only the indi.;tinguisl~~hiliq
of the two
modes in medium 1 but also that I*", < v,,.
(vii) The phenolnenon of mode crossing observad in some cases ir p s d b l y due to
coupling of modes ar'ising due to spatial rnattuistion of cornkptionq.
hfODUl.ATED SURFACE WAVE STRUCTURES
11
5. Phase eonstant (/?)
The problem of the propagitioa or E , wave in spatially modulated surface wave structuresisfnrnlulated in thc form of Hill's eqoation, rhe solt!tinn of which yield5 the
following relation'
where the dcterlninanl
sin" zfi
and involves 0, (11
- (),I,2, . . .) where
Where the length of a cell L = , f ( b , d3)which can he determined from Fig. 2. The
variation of 0. (up to I Z -.;4) with respect to 8 is shown in Fig. 6 . The variation of 9 [O]
with 6 i s shown in Fig. 7.
6. Radial field distributions
The field components in tho two media ere given bys
Med 1:
b<p.=~
Fro. 6 (u). Variation of N,,with S. with
pnrameter.
Ep2= - Aq
Where
r'l
as
Ro. 6 (1)). Variaticnl of
parameter..
Nt
wit11
8. with, en
[ ll (k@) -b. u
) K, (k@)1i
KO@
(kg4
(Ill
x and x1 involve the Fourier gap coeflicients c. (P).
The field distributions (E: and Hgwithp) in both the media are shown in Fig. 8 an1
9. The following condusions are of internst.
(i) The radial field de:ay in the outside medium for modulated (6 > 0) strUctU:ei
is h t e r than for an unirorm (6 .- 0) slrucure. Hence rho surface wave is man
tigitly bound in the case of.modulated stnwtures with low modulation indo
compared Lo the case of an uniform stmcture.
(ii) The end-fire property of surface wava modulated atnrctures d e w s not only on
a proper choice of 6 but also of L. The fast that better end-fire property m n k
atlained by slow rate of variation of s, i.c,., c ( 2 ) is also supported by Billstram',
(iii) The higher modas de:ay faster in med. 1 than the lower order modes, since
4 , > k,,. The greater consontration of energy towards the interface lead
>
to
greater Joulean heat loss for higher or&r modes b n tho &st order mode.
Conrequently, the first order mode will be guided to a longer distance along
the stm-tnre. Heme for a finita length modlrlated structure, the end-fire radia.
lion will consist mainly of the first order mode.
&
MODULATED SURI:ACI: WAVE STRUCTURTS
7. Fourier gap coefficients
The Fourier gap cooficicicnts c, Ibr d f o r r n l y corrugated rod is gwen by3
where 1 is the length of a period and n : 0, :+ 1, 2. & 3 order of forward (-I-)and
backward (-) spaee harmonics. F,,(k,p)/F, (k&) takes into acco~!nt the natL!re of the
geld variation in the radial direction. For a particular valne of c,, -- f (b,,s/2). It is
found that over the region &s/2 -;. 0 to 0.4 c, is a very slowly varying fmstion of J.
Hence, in the ease of the spatially modulated structure, the variation of c, may be considered to be inappreciably small. Hence, c, (B) fbr rhe modulated structure may be
considered to ha the same as c, in the case of uniform stnr.cti!re withont introducing any
appreciable error, provided 8 is small.
l'hc histogram or
e,,,,, arc shown in
(i) Tor slructrtres with
for slructures with
(ii) All a*,
(it+
c"
cU
Fig. lo. I[ i a irbacrvetl ih;rt
- 1. 53 1, 1.408 sn3 1. 179. I.+, En 4 O)!c,, .; I whereas,
= 1.062, 1'293, 1, $23 and I . 144 i. , ' c,,,.
-
0) in tho caw ul' struoturcs with
f'(cm)
FIG.8 (a). H+ vs p for 4 = 1.293 in medium 1.
-
GO
.. 1.53 I . I .JOX :n~d1.179 are
8. Power flow in modulated strucfures
The power Row outside (P;)and inside ( I J : )the tl~od~!l;wl
~ncdiittnin the z-direction
are given by the k)lluwing ralniiotn"
1.0
MED I
t
Ez
0
0.1
0.4
"0.6
0.6
1.0
1.2
1.4
1.6
1.8
2
MED
where
I1
FIG.9 (a). H, us p for
eQ = 1,ObZ.
MOUULATEU SLIRFNC W A V E STRUCTURES
and
2
m
-
c;
[
"--Lm
-
2
f
iQ)
sin ii
m-m
I;,
]
( 1 -- h c w k)
L
(8)sin 0 i -
,
n--x
.
;Cii,cos u
8
n r .OO
cm (j3j
,,roO
q:
\in 0
I,
2
m
sin
i
wl)
$
i,,,
( p ) sin 0
-02
(4) sin 0
iR
2
cly%
1
-l
c n 0-
r.-..WJ
I
Since 8, is a slowly varying rmction or z, it doei not vary signiiicnntly over a period
L. Hence, c,(j3) docs not vary significantly over a period L . The order of smallness
of variation may be estimated as (WKUJ approximation).
4
q
and hence
c.
>
d
V
"4
_ c,L" ,"here +:-d"dp- &'&*dj3
dc
-$
can be replaced by 5' which is equivalent to replaciny $
:
L
by
2
'he relative intewitie~of spatial hhnrcnonirsI,,,
(nf O)/fefor structures with different
valw of @ and 6 shown in Fig. I I in the form of his!ograms lead to the following
conclt!sions.
(i)
f:. (nZO)/f, c 1 for all structures.
(ii) For all structrrrm with anyvalue of A$.., >,f,,,but f,.(n,
the corresponding values of n.
-
I)
sf-,
(n s
1) for
(lii) For a structure with s particular value of 8. S has very little infilrence on
f* (n# 0)if. within the rnnge of S 0.02 to 6 = 0.1,
Frc. 11. Hi;Lo~amsof 1:-c relative iniensiticc ofspaiiH1 h a m n i t s f :i- (rzf-O)[fw
or the I - n t h Or /'" 'I"",; a i l l 1 r c i j r e ~ ~tol tT and I n -'I, \vllere P I p! .;. pJ,
presented i n Fig. I ? ~;lu'ii ili:ir
(i) P:/PzT = I'( ') cull1 C L ! ~tLi:lWid\ im t l ~ c. i : l ~ w c dvaltreb oi' /?, i.r.. t l ~ cs t a b i l i t y
The
I--
oi' modei.
(ii) P:/PT tcnil.; to iwrc:~v;i: :IS , i -+ 1). 111~ l l i ~ \\o[.di
i.
I I I C Z < I ~ \ C ~ ~ ! L ' I ~ofC Cmndulating ~ h \.r,!cil~ru
c
i.. 10 .~t!hiiain moi-e p m c r io f l o ~in ihv iiiodr~iatzdmcdirm,
thus ensuring a marc ii!:htiy h i ~ t ~ nsdi ! ~ f x c w:i\e.
P?/PF v* ?rz/l, for
I \ t roof. - - - 2nd T L ~ .
Fro. 12 (i,)
--
r*
! .??t.
1'1u. I3 (171.
klb :old i,.h 1 s l n z ! for
.- 2nd root.
l ~ root,
t
co
-
1 .5?3.
where
25
Fro. 12 (it).
T. S. RCTEhllNL AND S. 1;. OIIATTER.IEE
X,h vs 2
4 for so
-
1.062 (I st root)
MOI>ULATEI)SURFACE WAVE STRWCTLJRES
27
10. Attenuation constant
$
PI
(k&l -1.
K: (Ib) L
1 lk2a)
kz)
KL(@I]
and
(i)
As 6 -t 0, i.e., &, + &, i.e., the modulated structure tends towards a miform
structure, a is driermined with the understanding thaL the transverse propagation constants kcL and k, and the axial propagation constant have the values corresponding
to the values of a structure with uniform spacing but with the same values or h and a.
-
(ii) As b + u but (b
a) # 0, i ~ . for
, small depth ol" corrugation, the terms illvolvins
(b - a)lk, in (16) can be neglected and hence a reduces to
28
T. S. RUKMlNl AXD S. K . CIIATTBRIEE
I
0
0
1
2
3
4
6
5
7
8
--
P ~ / L
Frc. 1 3 (b).
(I)
a
vs Zm/L for d~ffcient8s for ra = 1.513.
The variation of a vs 2nzIL (Fig. 13) shows that a,,, - a,,,,, is much less for lower
valLles of 8 = 0-02, 0.04 than for higher ~ a l u e sof 6.
(ii) a,,,, for all values of 6 is attained at 2nzlL = n.
(iii) a,,(6+ 0) (Table I) calculated by considering values of a a t 1116th interval
over one period is greater than e (6 = 0).
(iv) a,, (6 # 0) > a (6 = 0) is justified since
5Pz* ( 8 = 0 ) > $ $ % ( 6 > 0 ) , ..The outside mediem, being air, islo?sless. Hence the loss for the guided wave
is contribnted by only the. physical properties of the moduiatec. medium.
(v) For shallow grooved strtrcturesa vr 2m/L (Fig. 14) shows that (Table 11) a,,
;s higher compared to that given in Table I,
-
0
MODULATEV SURbAC6 W A V E STRUCTURES
Table 11
a,,
- a for shallow grooved structures
io = 1
--
0662
6
0.02
0.04
0.06
0-08
0.1
-~e-= 1,523
o,, - R
0.7931
0.7927
0.7915
0.7901
0,7891
Since the cormgated line approaches Solnmerfeld line as b
a, ohntic dissipation giving
rise to Jonle heat loss is mainly responsible (radiation loss being l~eglectedin both the
cases) for supporting surface waves, i t is reasonable to expect that
a(shallow grooved line)
> u (b .;
a)
b-a
11. Surfate impedince
The real and iinaginary p a r k of sarface impedance are givcn in Lhe followiilg dimensionl e s form"
0
.
4
0
1
1
3
2
I
#
4
5
ZTTZ/L
RG. 14 (b), a vs ?m/L for diifeccnt 6s for
and
2nh X,
,
-.
wherc k, = a,
=
- [qb arc tan
- ib,
and 6,
=
U,b
"E
_d
6
7
8
1.523.
L?)~I
+ b,b lo O 89 (a: b' + i;
(186)
9
a,
The radial lield de:ay and atlcnuation depend respectively on X: and R, which a.re
intllen<ed by 6. Figure 15 provides design data for developing a strwture characterised by a rapid radial Reld d e a y and with minimnm attenuation. It is desirable to
have hi.gl.l pos!tive sarface reactance findi~xtive)andlow surra~eres:stance X, = 123
R, Ke(ZJ are /'(a, 8) and losation (2) along the structure, hcnce ~ h average
c
values
are plotted in Pig. 15. The average values have been calculated bq coneidcring thc value?
of the paTanleterS at 1/16 interval over a whole period,
-
31
MODULATED SURFACE WAVE STRUCTURES
FIG.15 (a-d) (a) R,(,,,lZo vs 8 for 4 = 1.062 and ao = 1.523. (6) a,, nkm vs 8 for an 1.523
and so = 1.062; (c) X,,,,,/Zo vs 6 for 8 = 1.062 and to = 1.523; (d) Z,(,,,/Z, vs 8 for
@ = 1.062.
5
We conclude from Fig. 15 that
(i)
For structure with
€0 = 1.062, the design shonld be for 6 = 0.08 or 0 . 1 so that
high rate of radial field decay with low attenuation can be obtained.
(ii) For strircture with r = 1.062, the design should be for 6 = 0.06 or 0.1.
12. Measurement of radial field decay
Radial field decay for several modulated and a uniform corrugated structmes was
measured with a monopole probe (Fig. 16) and the results are compared with theory
(Fig. 17). In the second and fourth illustrations of Fig. 17, the experimental (00) and
theoretical curve coincide for 6 = 0, whereas, for 6 = 0'04 there is a slight divergence
between the two curves.
We conclude, that for modulated structures, the rate of radial field decay is faster
than for uniform structure.
32
T. S. RUKMINI AND
S.
K. C H A ~ E R I E B
FIG.16. Photogiaph
_
.
of the structure.
13. Stability of the surface wave mode
The stability charts (Fig. 18) for structures wilh eo = 1,062 for S = 0'08 and 0.25 have
been prepared with ,5 as a parameter by irsing the equation (10) with the aid of a 360
IBM computer. (The Flow Chart is not reported).
The stable solutions corresponding to modulated propagating waves are associated
with the 8-values bounded a x o r d u g to the following inequality relation
The regions (hatched) which do not satisfy (20) correspond to the unstable solutions
of (LO). The unstable solutions are asso-iated with the non-propagating or damped
waves. The values of jare real within the shaded regions (unshaded), whereas they are
complex within the unstable regions. As 8 = f (&,8) and c0 = f (s, b, a) and 6
determines the propagating or non-propagating nature of waves, the stability charts
which determine k, can be used for the design of proper modulated structures which
support strongly bound surfase waves.
The location of the mode supported by the experimental structure en = 1.062, 8 =
0.08 is shown in Fig. 18 (a). The experimental point @lies within the unshaded regm
,5 = 3 and j= 4, thus indicating that the mode supported by the structure is stable.
It is found that the modes snp~ortedby other experimental structures with e , = 1.062
and S = 0.06, 0.04 and 0.1 also lie in the stable regions of stability charts which are
not reported.
Figure 18 (b) shows stability diagrams for the strmture with so = 1.062 and S = 0.25.
Similarly, the stability diagrams have also been drawn for other structnres with b = 0'08
and 0.25 for which no experimental structures were made.
MODULATED SURFACE WAVI? STRUCTURES
The following remarks regarding the stability charts are or interest.
(i) The curves separating the stable and unstable regions in some cases cross over
each other, for example, near the points (I, = 440, 480, 580, eic. (Fig. 18 (a))
where
tii) The area of the stable regions increase as the valw for /Iincreases. There is
also a tendency for the separation to inxease between two adjacent cross-over
points as p increases. As a consequence of increase in p, the waves become
more strongly bound. Hence, it may be said, that the structure for which the
nrmis no st strongly bound will exhibit the largest area of stable region with
widely separated cross-over points in the stability chart. Hence, the usefulness
the stability chart for desigoing a space modulated structure is obvious.
01'
14. Basic assumptions and their limitations
The iheorotlcal der~vat~on
of 11 for space modulated str~tctures1s based on the follow~ng
assumption6.
(i) The nonuniforn~lycorrugated metallic structure is considered as a conductor
coaled with a dielectric whose dielectric coostant varies cosinusoidally in the
direction of propagation.
1. This helps in truncating Iheseries for 0,
(ii) 8 :
in 9[01 and hence for B.
(11
=;
1,2,
. . .) which k involved
(ii) For deriving the profile equation for the design of a practical structure, small
and large argumenl approximations Tor the Bessel fnnclions have been introduced from practical considerations. This may impose some limitations o n the
accuracy of the resvlts. But considerjng the accuracy which can be attained,
in practice, in fabricating the structure it is considered that the argument
approximations are valid for all practical purposes.
(iv) The ain~ulateddielectric medium is considered lossless in deriving 8.
(c) I;n deriving the magnitudes of the spatial harmonics, 111and hence the Fourier
gap coefficients emu)
is considered to be a slowly vacyu~gfunction over a period
of tho modulatiol~cycle, which leads to sinlplification of the expression forf, (/I).
(vi) The coating thickness of the equivalent dielectric is considered tke sane as (b - 0).
(vii) In deriving Po,the field outside the slructurc is considered to be pure svrfaie
wave iield and the effect of diffraction is ignored. The assumption, however,
is justified in view of ihe higher value of /3, and hence a lower value for v,.
This is conferred by the experimental measurement of radial field decay.
15. Conclusious
(1)
Cosinusoidal lnodulation of spacing belwecu discs of circular cylindrical metallic
corrugated structures lead to more strongly bound stable surface waves.
(ii) Corrugated metallic structures may be considered as equivalent to a conductor
coated with equivalent dielectric whose dielectric constant is a function of the
spacing between discs, radius of ihe discs and radius of the central supporlinp
rod.
36
T, S. RUKMINI AND S. K . CHATTERJEE
(iii) Th'e corrugated modulated region provides inductive reactance which helps in
snmorting surface waves.
(iv) The at:enuation constant decreases with increase in the value of 6.
(.v) The power flow outside the modulated structure with respect to the total poher
flow dexcases with increasing 6.
... ...-. . . .
15. Acknowledgement
.. .
.-
.-
. < -
l t 1; a ?:ea; !re LO a-knowledge the invalnable suggestions by Dr. James R. Wait
(ESSA~;;'NOAA),'Monitor PL-480 ~ r o ~ r a m c n(Contract-E-262-69
e
(N), i969) and the
fi&n(tal &i.is?an-,e at the initial stages of the problem by th;e PL-480 authorities.
. O x grateful thanks are dso to Prof. -(Mrs.). R. Ehatterjee, Chairman, Department
of E!e-trkal Communi.cation Engineering, for encouragement and -helpful discussions.
...
,
. .
.
.
References
!.
.
CHATTWEE,
S. K.,
RUKMINI,
T. S. A N D ,
CHATTERIEE,
R.
2. CHATTLRJEB,
S. K.
RUKMINI;
T: S.
.
AND
-
.
Propagation of E, waye in modulated artificial dielectric media,
Indian Inst. Sci., 1977, 59 (A), 225-45.
..-.J.
Sirnulatioil of cosinusoidally modulated dielectric medium by
artificial dielectric at microwave frequelkiks, J: Indian Inst Sci.,
1978, 60 (A), 47-64.
3. C H A T ~ B WS.E ~K.;
,
RUK~WI,
T. S. AND
Surface waves in a nonunlformly conugated cylindrical metdlic
StNCtUre, J . Indiun Inst. Sci.,1978, 60(A), 119-134.
CHATTKRIEE, R.
d;. BILLSTR~M,
0.
E.M. Theow and ~nknnas, Editod by E. C. Jordan, Pergarnon
Press, London, 1963, p . 875.