On the use of SDA for the analysis of boxed planar lines with complex media
Abstract
This paper discusses the conditions under which the spectral-domain approach (SDA) can be applied to the analysis of boxed planar lines when complex materials (anisotropic dielectrics, ferrites, magnetoplasmons, chiral media, and so on) are used as substrates. It will be shown that whereas SDA can always be efficiently applied to study laterally open structures, the simultaneous presence of lateral boundary conditions and nonisotropic materials requires further study. Thus, the symmetry properties of the constitutive dyadics that makes possible a rigorous application of the SDA to those kinds of structures will be reported in this paper.
Full text
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 7, JULY 2001 1365
On he Use o SDA o he Analysis o Boxed Plana Lines
wi h Complex Media
Gonzalo Plaza, F ancisco Mesa, and F ancisco Medina
Abs ac —This pape discusses he condi ions unde which he spec-
al-domain app oach (SDA) can be applied o he analysis o boxed plana
lines when complex ma e ials (aniso opic dielec ics, e i es, magne o-
plasmons, chi al media, and so on) a e used as subs a es. I will be shown
ha whe eas SDA can always be e icien ly applied o s udy la e ally open
s uc u es, he simul aneous p esence o la e al bounda y condi ions and
noniso opic ma e ials equi es u he s udy. Thus, he symme y p ope -
ies o he cons i u i e dyadics ha makes possible a igo ous applica ion
o he SDA o hose kinds o s uc u es will be epo ed in his pape .
Index Te ms—(Bi)aniso opic media, shielded plana lines, spec al do-
main.
I. INTRODUCTION
The echnological ad ances in he de elopmen o complex ma e-
ials has aised a heo e ical and p ac ical in e es in he analysis o
mic owa e ansmission lines and de ices wi h subs a e laye s o e y
gene al p ope ies. This esea ch opic is expec ed o lead o new mi-
c owa e de ices and/o o enhance he pe o manceo he cu en ly ex-
is ing ones. In his way, a g ea e o is also de o ed o s udy he p opa-
ga ion and/o adia ion cha ac e is ics o a numbe o plana s uc u es
whose laye s a e made o aniso opic, gy o opic, o bi(an)iso opic
ma e ials. Among he mos common echniques o deal wi h plana
s uc u es, hose based on he applica ion o Fou ie ans o m (FT)
echniques, such as he spec al-domain app oach (SDA), ha e p o en
o be e y e icien [1]. This me hod has been success ully ex ended
o deal wi h aniso opic and bi(an)iso opic subs a es in la e ally open
s uc u es [2], [3]. La e ally shielded con igu a ions ha ing ce ain pa -
icula aniso opic media ha e also been p ope ly ea ed in he li e a-
u e [4]–[6]. Howe e , he di ec applica ion o he SDA exp essions
(o iginally de eloped o la e ally open s uc u es) o he analysis o
boxed s uc u es needs a ca e ul examina ion when noniso opic ma e-
ials a e in ol ed [7]–[10]. The inapp op ia e use o he SDA o s udy,
o example, boxed guiding s uc u es con aining longi udinally mag-
ne ized e i e/semiconduc o and/o chi al ma e ials, could b ing con-
cep ual e o s ha would yield inaccu a e nume ical esul s. Thus, he
p esence o la e al elec ic walls (EWs) and/o magne ic walls (MWs)
equi es deepe examina ion. Al hough his poin has been add essed
in he li e a u e o ce ain speci ic cases in a ious con ex s [11]–[14],
i seems ha some p ac i ione s o he SDA a e no comple ely awa e
o his ac , pe haps because a comp ehensi e discussion on his opic
in he ame o he SDA has no ye been epo ed. Thus, he main goal
o his pape is o cla i y he condi ions unde which he SDA is sui -
able o he analysis o boxed plana lines including a bi a y complex
linea media as subs a es. A e add essing he na u e o he p oblem
wi h simple examples, he s udy will p o ide c i e ia ( ela ed o he
symme y p ope ies o he cons i u i e pa ame e s o he ma e ials) o
know when he SDA is easible o analyze he abo e-men ioned ype
o guiding s uc u es.
Manusc ip ecei ed No embe 10, 1999; e ised Sep embe 27, 2000. This
wo k was suppo ed by he Spanish Comision In e minis e ial de Ciencia y Tec-
nologia unde P ojec TIC98-0630.
G.PlazaandF.Mesaa ewi h heDepa men o AppliedPhysicsI,Uni e si y
o Se ille, 41012 Se ille Spain (e-mail: [email p o ec ed]).
F. Medina is wi h he Depa men o Elec onics and Elec omagne ism,
School o Physics, Uni e si y o Se ille, 41012 Se ille Spain.
Publishe I em Iden i ie S 0018-9480(01)05046-3.
Fig. 1. C oss sec ion o a plana laye ed line. The cons i u i e dyadics o each
laye a e assumed o be homogeneous o
0
<x<a
.
II. APPLICATION OF SDA TO PLANAR LINES
Al hough he SDA is ac ually a well-known and widely used ech-
nique, i will be b ie ly ou lined he e o su ey he possible d awbacks
appea ing when he echnique is applied o plana lines wi h la e al
bounda y condi ions (BC) and noniso opic ma e ials. Fig. 1 shows he
c oss sec ion o a gene ic laye ed plana line wi h ec angula BCs:
EWs, MWs, o pe iodic walls (PWs) exis a
x
=0
and
x
=
a
. (The
absence o hese walls leads o la e ally open lines.) In gene al, he
subs a e laye s a e homogeneous linea ma e ials whose cons i u i e
pa ame e s a e gi en by he ollowing linea dyadics:
D
=
1
E
+
1
H
B
=
1
E
+
1
H
D
B
=[
0
]
1
E
H
(1)
which accoun o he simples iso opic dielec ic o he mos complex
bianiso opic ma e ial.
Assuming a ield dependence o he ype
A
(
;
)=
A
(
x; y
)exp[
0
j
(
k
z
z
0
!
)]
, Maxwell cu l equa ions can be
w i en as
0R
0
0
R1
E
H
=
j!
B
D
=
j!
1
E
H
(2)
whe e
R
=(
^
x
^
y
0
^
y
^
x
)
jk
z
+(
^
y
^
z
0
^
z
^
y
)
@=@x
+(
^
x
^
z
0
^
z
^
x
)
@=@y
is he cu l ope a o . Elimina ing now he
y
-componen s o he
ields, he ollowing ma ix i s -o de pa ial di e en ial equa ion
[subjec o he app op ia e BCs
(
x; y
)
] can be o mally ound o
X
=[
E
x
;E
z
;H
x
;H
y
]
[2]:
D
(
@=@x; @=@y; k
z
;!
) [
X
]=[
Q
(
!
)][
X
]
BC
's
(
x; y
)
(3)
whe e
[
D
(
1
)]
is he esul an i s -o de ma ix di e en ial ope a o and
[
Q
(
!
)]
is a ma ix accoun ing o he laye ed medium. The s anda d
applica ion o he SDA [1] o sol e o he abo e pa ial di e en ial
equa ion consis s basically o applying a combina ion o G een’s unc-
ion me hods and an app op ia e spec al ep esen a ion.
Nex , le us examine he di e en BCs in he la e al
x
-di ec ion in
connec ion wi h he applica ion o FTs.
A. La e ally Open Lines
This case shows an open domain in he la e al di ec ion since he e
a e no BCs in he
x
-di ec ion. Func ions o he ype
exp(
0
jk
x
x
)
a e
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1366 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 7, JULY 2001
Fig. 2. La e ally pe iodic line o pe iod
a
.
imp ope eigen unc ions o he p oblem, which leads o he ollowing
in eg al FT:
X
(
x
)= 1
2
1
01
~
X
(
k
x
)exp(
0
jk
x
x
)
dk
x
(4)
~
X
(
k
x
)=
1
01
X
(
x
)exp(
jk
x
x
)
dx
(5)
as he na u al spec al ep esen a ion [16].
A e he applica ion o he in eg al FT, he ollowing ma ix i s -
o de di e en ial equa ion has o be sol ed:
D
(
@=@y; k
x
;k
z
;!
)~
X
=[
Q
(
!
)] ~
X
BC
's
(
y
)
:
(6)
Applying he G een’s unc ion me hod along he no mal
y
-di ec ion
can now sol e o he abo e di e en ial equa ion.
I is e y impo an o no e ha in w i ing (6), i has made use o he
ac ha he subs a e laye s a e homogoneous along he whole la e al
x
-di ec ion (
01
<x<
1
), namely, he cons i u i e dyadics o he
p oblem do no depend on
x
. This ac is ele an when aking he FT
o
D
- and
B
- ields. Thus, i
D
(
x
)=
1
E
(
x
)+
1
H
(
x
)
(7)
hen he co esponding spec al coun e pa is gi en by
~
D
(
k
x
)=
1
~
E
(
k
x
)+
1
~
H
(
k
x
)
(8)
i.e., he e exis s a simple linea ela ion be ween he spec al compo-
nen s o he ields.
B. La e ally Pe iodic Lines
Since he s uc u e unde s udy (as ha shown in Fig. 2) is ully pe-
iodic, a se ies FT
X
(
x
)=1
a
n
~
X
n
exp(
0
jk
x; n
x
)
;k
x; n
=
n
2
a
(9)
can be applied o sol e (3). As in he p e ious case, due o he homo-
genei y o he laye ed medium o all he
x
- ange, i can be w i en ha
~
D
n
=
1
~
E
n
+
1
~
H
n
(10)
and he ollowing o dina y ma ix di e en ial equa ion is eached:
D
(
@=@y; k
x; n
;k
z
;!
)~
X
n
=[
Q
(
!
)] ~
X
n
BC
's
(
y
)
:
(11)
The la e ally pe iodic case is hen equi alen o he la e ally open
case, excep ha he con inuous Fou ie a iable (
k
x
) is now eplaced
by he disc e e Fou ie a iable (
k
x; n
; namely, he ha monics in ol ed
in he Fou ie expansion o he sou ces). I should be highligh ed ha ,
o he la e ally open and pe iodic cases, he SDA could be e icien ly
applied because he subs a e laye s o he s uc u es we e homoge-
neous along he la e al di ec ion.
C. La e ally Shielded Lines
I FT echniques a e wished o be applied o he shielded s uc u e
o Fig. 1, i is i s necessa y o o m an equi alen pe iodic line (EPL),
ha ing exac ly he same p opaga ion cha ac e is ics. The uni cell o
basis-pe iod line (BPL) de ining such an EPL is gene a ed by using
he well-known echnique o images o eplace he EWs/MWs by a
se ies o equi alen sou ces and media. I should be no iced ha he
imaging o sou ces implies he p ope e lec ion o all he o iginal
sou ces, namely, bo h he ee/imposed and induced sou ces (pola iza-
ion cha ges, magne iza ion cu en s, e c.) by he la e al walls. Since
he e ec o he induced cha ges is accoun ed o by he dyadic cons i-
u i e pa ame e s, he imaging o he induced sou ces will be e lec ed
by he imaged cons i u i e dyadics. Thus, he whole imaging p ocedu e
can be seen as a e lec ion ope a ion o bo h he o iginal ee/imposed
sou ces and he cons i u i e dyadics o he media. Al hough, o he
iso opic case, he imaged medium coincides wi h he o iginal one, his
does no happen, in gene al, o noniso opic media. (This la e ac is
wha seems o be ob ia ed by some SDA p ac i ione s.)
A e lec ion by a plane loca ed a
x
=0
is accoun ed o by he
ollowing linea ope a o :
R
=
0
^
x
^
x
+^
y
^
y
+^
z
^
z
=
R
0
1
:
(12)
I he symme y plane is an EW, hen
R
EW
=
0
R
and
R
MW
=
R
o an MW. The p ope applica ion o he abo e ope a o o he o iginal
ee/imposed sou ces will gi e he imaged sou ces. Ob aining he im-
aged cons i u i e dyadics will be illus a ed, making use o how ield
ec o s/pseudo ec o s e lec .
The imaged ields a e gi en by
E
R
D
R
=
R
(EW
=
MW)
1
E
D
(13)
H
R
B
R
=
0
R
(EW
=
MW)
1
H
B
:
(14)
Takingin o accoun he gene al cons i u i eequa ionsgi enin(1),a e
s aigh o wa d algeb a, he ollowing cons i u i e ela ions a e ound
o he e lec ed ields:
D
R
=(
R
1
1
R
0
1
)
1
E
R
+(
0
R
1
1
R
0
1
)
1
H
R
(15)
B
R
=(
R
1
1
R
0
1
)
1
H
R
+(
0
R
1
1
R
0
1
)
1
E
R
(16)
which implies ha he e lec ed cons i u i e dyadics a e hen gi en by
R
R
=
R
1
1
R
0
1
(17)
R
R
=
0
R
1
1
R
0
1
:
(18)
Ca ying ou he app op ia e e lec ion ope a ions, Fig. 3 shows he
BPLs o h ee di e en common si ua ions appea ing in p ac ice. In
some cases, he e lec ed dyadics (
0
R
in Fig. 3) can coincide wi h he
o iginal ones (
0
), hus, gi ing place o a BPL ha is
x
-homogeneous in
i s whole pe iod o de ini ion. Unless his la e si ua ion is ound, he
applica ion o he disc e e FT o sol e o (3) is no ad an ageous [13].
In o de o cla i y he abo e poin , he ec angula wa eguide loaded
wi halongi udinalmagne ized e i eshowninFig.4(a)willbeconsid-
e ed. Taking in o accoun ha , in he p esen case, he ex e nal biasing
magne ic ield changes i s sign unde e lec ion in he conduc ing plane
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IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 7, JULY 2001 1367
Fig. 3. O iginal lines (le -hand side) and BPLs o he EPL ( igh -hand side)
esul ing om he e lec ions by he la e al BCs o h ee di e en si ua ions.
(a) MWs. (b) EWs. (c) E/MWs.
0
ep esen s any o he cons i u i e dyadics.
Fig. 4. (a) Shielded ec angula wa eguide illed by a longi udinally
magne ized e i e. (b) BPL o he EPL ac ually analyzed by he SDA.
[12], [17] (which is equi alen o e lec he Polde enso ), he co e-
sponding BPL o he EPL is ha shown in Fig. 4(b). Since he dyadic
pe meabili y o he BPL is gi en by
(BPL)
(
x
)=
;
0
<x<a
T
;a<x<
2
a
(19)
i is ound ha he spec al coun e pa o
B
(
x
)=
(BPL)
(
x
)
1
H
(
x
)
(20)
is he ollowing con olu ion p oduc :
~
B
n
=
1
m
=
01
(BPL)
n
0
m
1
~
H
m
:
(21)
Thus, all he spec al componen s o
H
a e ela ed o all he spec al
componen s o
B
. This gi es ise o an in e dependence among all he
spec al componen s o he di e en ields ha would lead o he ol-
lowing sys em o di e en ial equa ions:
D
(
@=@y; k
x; n
;k
z
;!
)~
X
n
=
1
m
=
01
[
Q
n
0
m
(
!
)] ~
X
m
BC
's
(
y
)
:
(22)
Ce ainly, he igo ous solu ion o he abo e sys em o coupled di e -
en ial equa ions is an imp ac icable ask.
In conclusion, he SDA can always be applied o la e ally shielded
lines, al hough i is only sui able o hose cases sa is ying
0
R
=
0
,
namely, when he medium o he BPL is homogeneous along he la e al
di ec ion.
III. CONDITION FOR HOMOGENEITY OF THE BPL
A he ligh o (17) and (18), he condi ion o he exis ence o he
equi ed e lec ion symme y
0
R
=
0
can be exp essed as
R
1
=
1
R
(23)
R
1
=
0
1
R
:
(24)
Thus, he condi ions o keep unchanged he medium a e he co e-
sponding e lec ion impose ha he cons i u i e dyadics mus ha e he
ollowing gene ic explici o m:
=
xx
^
x
^
x
+
yy
^
y
^
y
+
yz
^
y
^
z
+
zy
^
z
^
y
+
zz
^
z
^
z
(25)
=
xx
^
x
^
x
+
yy
^
y
^
y
+
yz
^
y
^
z
+
zy
^
z
^
y
+
zz
^
z
^
z
(26)
=
xy
^
x
^
y
+
xz
^
x
^
z
+
yx
^
y
^
x
+
zx
^
z
^
x
(27)
=
xy
^
x
^
y
+
xz
^
x
^
z
+
yx
^
y
^
x
+
zx
^
z
^
x
:
(28)
The abo e exp essions clea ly es ic he kind o subs a e laye s
ha can be p esen in hose boxed s uc u es o be analyzed by means
o he SDA. Speci ically, (25) and (26) es ablish ha he SDA will be
sui able o he analysis o hose lines ha ing dielec ic/magne ic sub-
s a es whose pe mi i i y/pe meabili y dyadics show a p incipal axis
no mal o he la e al bounda y walls. This condi ion will be sa is ied,
o example, by iso opic ma e ials, uni/biaxial dielec ics ha ing one
o he p incipal di ec ions along he
x
-di ec ion, and plasmons/ e i es
biasedbya magne ic ielddi ec ed in he
x
-di ec ion(
H
0
=
H
0
^
x
). Fo
hecaseo bi(ani)iso opic ma e ials(
6
=0
,
6
=0
),(27) and(28)s a e
ha he SDA canno be igo ously applied o boxed s uc u es ha ing,
o ins ance, biiso opic ma e ials and chi o e i es. In ac , (27) and
(28) p eclude he use o he SDA o he analysis o boxed lines wi h
almos any kind o bianiso opic media.
I he abo e conside a ions we e igno ed and he SDA was used
o s udy boxed s uc u es no ul illing he equi emen s gi en in his
pape , he nume ical esul s ob ained could be app oxima ely co ec
p o ided he box wid h is la ge in compa ison wi h he o he dimen-
sions o he s uc u e. Ac ually, nume ical se ies could be conside ed
in such cases as app oxima ions o he Fou ie in eg als appea ing in
he analysis o open lines. The esul s o hese closed s uc u es should
a he be conside ed as app oxima ions o he co esponding la e ally
open s uc u es. I he in luence o he la e al shielding could no be ne-
glec ed, he esul s ob ained wi h codes no accoun ing o he heo y
in his pape a e expec ed o be inaccu a e.
IV. CONCLUSION
This pape has s udied he condi ions unde which FT echniques
can be p ope ly applied o s udy boxed plana lines in ol ing laye s
o complex linea media. Since he applica ion o he SDA o s udy
he p opaga ion p ope ies o boxed lines is educed du ing he anal-
ysis o a uni -cell line o an EPL, only he p esence o homogeneous
subs a e laye s in he uni -cell line will allow o a sui able applica-
ion o he echnique. The homogeneous na u e o he subs a e laye s
o he BPL is de e mined by he e lec ion symme y p ope ies o he
media and, hus, he condi ions o be sa is ied by a gene al linea bian-
iso opic ma e ial ha e been s udied. Finally, i has been concluded ha
he SDA can only be p ope ly applied o he analysis o boxed plana
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1368 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 49, NO. 7, JULY 2001
lines wi h complex media whose cons i u i e dyadics a e o he ype
gi en in (25)–(28). These exp essions p eclude he use o he s anda d
SDA o igo ously s udy boxed plana lines wi h dielec ic/magne ic
ma e ials ha ing a p incipal axis no di ec ed along he
x
-di ec ion bi-
iso opic ma e ials as well as almos any ype o bianiso opic ma e ial.
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