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
0018–9480/01$10.00 © 2001 IEEE
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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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