S ong spa ial dispe sion in wi e media in he e y la ge wa eleng h limi
P. A. Belo ,1,2 R. Ma que
´s,3S. I. Maslo ski,1I. S. Ne edo ,1,4 M. Sil ei inha,5C. R. Simo ski,1,2 and S. A. T e yako 1
1Radio Labo a o y, Helsinki Uni e si y o Technology, P.O. Box 3000, FIN-02015 HUT, Finland
2Physics Depa men , S . Pe e sbu g Ins i u e o Fine Mechanics and Op ics, Sablinskaya 14, 197101 S . Pe e sbu g, Russia
3Depa men o Elec onics and Elec omagne ism, Facul ad de Fı
´sica, Uni e si y o Se illa,
A enida Reina Me cedes s/n, 41012 Se illa, Spain
4Ins i u e o Radio Enginee ing and Elec onics, Russian Academy o Science, Zelyonaya 38, 410019 Sa a o , Russia
5Ins i u o Supe io Te
´cnico, Ins i u o de Telecomunicac¸o
˜
es, A enida Ro isco Pais, 1049-001 Lisboa, Po ugal
共Recei ed 27 No embe 2002; published 25 Ma ch 2003兲
I is ound ha he e exis composi e media ha exhibi s ong spa ial dispe sion e en in he e y la ge
wa eleng h limi . This ollows om he s udy o la ices o ideally conduc ing pa allel hin wi es 共wi e media兲.
In ac , ou analysis e eals ha he desc ip ion o his medium by means o a local dispe si e uniaxial
dielec ic enso is no comple e, leading o unphysical esul s o he p opaga ion o elec omagne ic wa es a
any equencies. Since nonlocal cons i u i e ela ions ha e been usually conside ed in he pas as a second-
o de app oxima ion, meaning ul in he sho -wa eleng h limi , he a o emen ioned esul p esen s a ele an
heo e ical in e es . In addi ion, since such wi e media ha e been ecen ly used as a cons i uen o some disc e e
a i icial media 共o me ama e ials兲, he epo ed esul s open he ques ion o he ele ance o he spa ial
dispe sion in he cha ac e iza ion o hese a i icial media.
DOI: 10.1103/PhysRe B.67.113103 PACS numbe 共s兲: 78.20.Ci, 41.20.Jb, 42.70.Qs, 78.70.Gq
Causali y imposes ha all ma e ial media mus be dispe -
si e. In mos cases his beha io esul s in local dispe si e
cons i u i e ela ions, i.e., in equency-dependen cons i u-
i e pe mi i i y and pe meabili y enso s. Nonlocal dispe -
si e beha io 共i.e., spa ial dispe sion兲, which esul s in con-
s i u i e ope a o s depending also on he spa ial de i a i es
o he mean ields 共o , o plane elec omagne ic wa es, on
he wa e- ec o componen s兲, is usually conside ed as a
small e ec , meaning ul in he sho -wa eleng h limi . Spe-
ci ically, spa ial dispe sion will always appea when he
highe -o de e ms in he se ies expansion o he cons i u i e
pa ame e s in powe se ies o he dimensionless pa ame e
a/␭(ais he la ice cons an o he c ys al and ␭ he wa e-
leng h inside he medium兲a e no neglec ed.1Thus, i is
usually assumed ha nonlocal dispe si e ea ions a e only
meaning ul when ␭app oaches a. The usually weak na u al
op ical ac i i y o some ma e ials is a well-known example
o he applica ion o his p inciple.1When such p inciple is
ansla ed o he analysis o disc e e a i icial media, also
called me ama e ials, i would imply ha nonlocal dispe si e
cons i u i e ela ions a e only expec ed o be a small
e inemen o he local cons i u i e ela ions usually con-
side ed. Howe e , he e is a leas a coun e example o his
assump ion.
The pa allel wi e medium is a medium o med by a egu-
la la ice o ideally conduc ing wi es wi h small adii com-
pa ed o he la ice pe iods and he wa eleng h, see Fig. 1. I
has been known in mic owa e applica ions o a long ime2–4
as an a i icial dielec ic, also called odded medium and qua-
sis a ic models o he e ec i e pe mi i i y a e a ailable.5
Recen ly, some a en ion o wi e media has been paid also in
op ics 共e.g., Re s. 6,7兲and in he ealiza ion o le -handed
media8,9 as composi e media made om la ices o long con-
duc ing wi es and spli ing esona o s10–12 共see discussions
co esponding o ha in13兲. The elec omagne ic eponse o
he speci ic wi e medium shown in Fig. 1 is analyzed in
Re s. 3,7 ollowing di e en app oaches. Bo h analysis a e
ca ied ou o wa e p opaga ion pe pendicula o he wi es
and show ha , o elec ic- ield pola iza ion along he wi es,
he medium is cha ac e ized 共i a/␭,b/␭Ⰶ1) by a equency-
dependen e ec i e dielec ic cons an gi en by
␧⫽␧0
冉
1⫺k0
2
k2
冊
⫽␧0
冉
1⫺
␻
0
2
␻
2
冊
.共1兲
The cons an
␻
0共 he co esponding wa e numbe k0
⫽
␻
0
冑
␧0
␮
0) in Eq. 共1兲plays he ole o an equi alen
‘‘plasma equency.’’ Thus, his medium is o en called ‘‘a -
i icial plasma’’ since he ideal 共collisionless兲elec on plasma
is desc ibed by he same equi alen pa ame e . I he wi es
FIG. 1. The geome y o wi e media: A la ice o pa allel ide-
ally conduc ing hin wi es.
PHYSICAL REVIEW B 67, 113103 共2003兲
0163-1829/2003/67共11兲/113103共4兲/$20.00 ©2003 The Ame ican Physical Socie y67 113103-1
a e assumed o be e y hin, so ha hei pola iza ion in he
di ec ion o hogonal o he wi es can be neglec ed, he e ec-
i e pe mi i i y o elec ic- ield pola iza ion o hogonal o
he wi es is ␧0.
The a o emen ioned analysis sugges s ha his wi e me-
dium could be modeled as a uniaxial dielec ic wi h he ol-
lowing local pe mi i i y dyadic:
␧
¯
¯
⫽␧z0z0⫹␧0共x0x0⫹y0y0兲,共2兲
whose pe mi i i y in he axial di ec ion, ␧, would be gi en
by Eq. 共1兲. Howe e , i will be shown in he ollowing ha
his nai e hypo hesis leads o unphysical esul s and mus be
subs i u ed by a nonlocal dispe si e ela ion. In ac , assum-
ing ha he medium can be desc ibed by he uniaxial dyadic
共2兲, he dispe sion equa ion o ex ao dina y plane wa es
(Ez⫽0) wi h he wa e ec o (qx,qy,qz)Tin his uniaxial
dielec ic eads14,15
␧0共qx
2⫹qy
2兲⫽␧共k2⫺qz
2兲,共3兲
whe e k⫽
␻
冑
␧0
␮
0is he phase cons an o he hos ma ix.
On he o he hand, hese ex ao dina y wa es co espond o
he well-known TM 共 o z) se o modes, allowed by he
in a iance o he bounda y condi ions along z. Thus, o any
ex ao dina y wa e a eling wi h a phase cons an qzalong
he zaxis, he Ez ield mus sa is y he Helmhol z equa ion
再
⳵
⳵
x2⫹
⳵
⳵
y2⫹共k2⫺qz
2兲
冎
Ez⫽0, 共4兲
wi h he bounda y condi ion Ez⫽0 on he wi es. I is clea
om his equa ion ha any ‘‘plane’’ex ao dina y wa e mus
sa is y
k共qx,qy,qz兲⫽
冑
k2共qx,qy,0兲⫹qz
2.共5兲
This las esul is incompa ible wi h Eq. 共1兲–共3兲, as can be
easily seen by subs i u ion o Eq. 共1兲in o Eq. 共3兲. Howe e ,
i we choose
␧共k,qz兲⫽␧0
冉
1⫺k0
2
k2⫺qz
2
冊
共6兲
ins ead o Eq. 共1兲, hen Eqs. 共2兲and 共3兲become compa ible
wi h Eq. 共5兲, gi ing he ollowing dispe sion equa ion o he
plane wa e:
q2⬅qx
2⫹qy
2⫹qz
2⫽k2⫺k0
2,共7兲
whe e we ha e assumed ha qz⫽k共 he case wi h qz⫽kwill
be analyzed a he end o his pape 兲. The abo e a ionale
sugges s ha he conside ed wi e media s ill can be desc ibed
by he pe mi i i y dyadic 共2兲, bu he axial pe mi i i y ␧
mus be a nonlocal pa ame e o he o m 共6兲. The con en-
ional exp ession 共1兲would be only a pa icula case o Eq.
共6兲, alid o wa e p opaga ion in he x-yplane.
The main di e ence be ween he local uniaxial model,
Eq. 共1兲, and he p oposed nonlocal model, Eq. 共6兲, o he
pa allel wi e medium is ha he nonlocal model p edic s a
s op band 共a equencies below
␻
0⫽k0/
冑
␧0
␮
0) o ex ao -
dina y wa es p opaga ing along any di ec ion in he media.
On he con a y, Eqs. 共1兲–共3兲p edic p opaga ion o ex ao -
dina y wa es a any equency p o ided qz⬎k⫽
␻
冑
␮
0␧0.
Thus, bo h models p edic quali a i ely e y di e en beha -
io s, e en nea he cu o plasma equency
␻
0whe e q2
→0共i.e., a/␭→0). Tha is, he nonlocali y o he p oposed
cons i u i e ela ions a ec s he elec omagne ic esponse o
he medium e en in he e y la ge wa eleng h limi , hus
being impo an o any alues o he a/␭ a io inside he
medium. O he ele an di e ences be ween he p edic ions
o bo h models will be de eloped along his pape .
The igo ous p oo o Eq. 共6兲is based on he local- ield
app oach which is desc ibed in de ail in Re . 16. In Re . 16
he low- equency s op band o he wi e medium has been
analyzed, as well as i s high- equency band-gap s uc u e.
This analysis e eals ha , in he hin wi e medium 共Fig. 1兲
and o qz⫽k, wo se s o modes can p opaga e: o dina y
共wi h Ez⫽0) and ex ao dina y 共wi h Ez⫽0) wa es. The
o dina y wa es do no in e ac wi h he wi es and p opaga e
in he hos media. Fo ex ao dina y wa es, an explici dis-
pe sion equa ion connec ing he wa e ec o q
⫽(qx,qy,qz)Twi h he wa e numbe o he hos iso opic
ma ix k⫽
␻
冑
␧0
␮
0has been de i ed in Re . 16. I can be
w i en as
1
␲
ln b
2
␲
0
⫹1
bkx
(0)
sinkx
(0)a
coskx
(0)a⫺cosqxa
⫹兺
n⫽0
冉
1
bkx
(n)
sinkx
(n)a
coskx
(n)a⫺cosqxa⫺1
2
␲
兩
n
兩
冊
⫽0.
共8兲
He e kx
(n)deno es he xcomponen o n h Floque mode
wa e ec o :
kx
(n)⫽⫺j
冑
冉
qy⫹2
␲
n
b
冊
2
⫹qz
2⫺k2,Re
兵
冑
共兲
其
⬎0. 共9兲
The o he no a ions a e clea om Fig. 1. Nume ical solu ion
o his dispe sion equa ion shows ha he e exis s a low-
equency s op band o all p opaga ion di ec ions 共excep
o he pa icula case o qz⫽k ha will be analyzed la e 兲.
Le us simpli y he dispe sion equa ion 共8兲 o he quasis a ic
case a,bⰆ
␲
/k. Using he Taylo expansion o sin(x) and
cos(x) unc ions o small a gumen s we ob ain Eq. 共7兲wi h
k0
2⫽2
␲
/s2
ln s
2
␲
0
⫹F共 兲
,共10兲
whe e s⫽
冑
ab, ⫽a/b, and
F共 兲⫽⫺ 1
2ln ⫹兺
n⫽1
⫹⬁
冉
co h共
␲
n 兲⫺1
n
冊
⫹
␲
6.共11兲
The e o e, we ha e shown he sui abili y o he sugges ed
app oach o he desc ip ion o he wi e medium, wi h k0
gi en by Eq. 共10兲. Pa ame e k0he e plays he ole o he
BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲
113103-2
wa e numbe co esponding o he plasma equency. Mo e
exac ly, i indica es he uppe edge o he low- equency s op
band. Na u ally, k0as a unc ion o wo la ice pe iods aand
bis a symme ic unc ion: k0(a,b)⫽k0(b,a). I means ha
unc ion F( ) has he ollowing p ope y: F(1/ )⫽F( ). Fo
he commonly used case o he squa e g id (a⫽b), F(1)
⫽0.5275. Exp ession 共10兲looks simila o he app oxima e
exp essions o he plasma equency de eloped ea lie in
Re s. 2–4,6, bu o hin wi es i is mo e accu a e and akes
in o accoun he geome y o he la ice. No ice ha he dis-
pe sion equa ion 共7兲 o ex ao dina y wa es is indi e en o
he di ec ion o he wi es’ axis: he wa e- ec o componen s
qx,qy,qzen e in o his equa ion comple ely symme ically.
I means ha , wi hin he low- equency s op band, he ex-
ao dina y wa e decays wi h he same decay ac o along all
di ec ions in space. The same can be said o p opaga ion in
he i s equency passband. This iso opy o he dispe sion
equa ion is a he su p ising since he medium is s ongly
aniso opic. Howe e , i can be shown om he e y unda-
men al ac s summa ized in Eqs. 共5兲and 共3兲by assuming, as
usual, ha Eq. 共1兲is alid o ex ao dina y wa es p opaga -
ing in he x-yplane.
I is possible o ansi Eq. 共6兲 om he spec al domain
(q,
␻
) o he physical domain ( , ). The ollowing nonlocal
ma e ial equa ion can be de i ed om Eq. 共6兲using he
double Fou ie ans o m:
D共x,y,z兲⫽␧0E共x,y,z兲⫹
␧0k0
2c
2z0
冕
⫺⬁
冕
z⫺c( ⫺ ⬘)
z⫹c( ⫺ ⬘)
⫻Ez共x,y,z⬘, ⬘兲dz⬘d ⬘,共12兲
whe e c⫽1/
冑
␧0
␮
0is he speed o ligh in he hos ma ix.
He e, he a ea o in eg a ion in he z- plane is he ligh cone
兩
z⫺z⬘
兩
⬍c( ⫺ ⬘). In o he wo ds, he ke nel in he Fou ie
con olu ion is u关c( ⫺ ⬘)⫺
兩
z⫺z⬘
兩
兴, whe e u(x) is he
Hea iside s ep unc ion. I means ha he poin (x,y,z) in-
side he wi e medium 共desc ibed as a dispe si e con inuum兲
a momen is a ec ed by he zcomponen s o elec ic ields
coming om he domain „x,y,z⫾c( ⫺ ⬘)…su ounding
共along he wi e axis兲 his poin du ing all he pas ime ( ⬘
⬍ ). This esul illus a es he consis ency o he epo ed
model om he ela i is ic s andpoin .
In he ollowing we will desc ibe some ele an e ec s in
he analyzed pa allel wi e medium, associa ed wi h he spa-
ial dispe sion. Re ac ion and e lec ion o plane wa es a a
plane in e ace show s ong di e ences be ween he local
and nonlocal models. Le us conside an in e ace be ween
an iso opic dielec ic wi h he pe mi i i y ␧1and a uniaxial
dielec ic wi h ␧desc ibed by Eq. 共1兲. The in e ace is in he
y-zplane and i is illumina ed by a plane wa e coming om
he iso opic dielec ic. The wa e ec o and elec ic- ield
ec o lie in x-zplane (qy⫽0,Ey⫽0). The incidence angle
o he plane wa e is
␪
.I ␧1⬎␧0,␧⬍0, and sin2(
␪
)
⬍␧0/␧1 he wa e will be comple ely e lec ed, bu o
sin2(
␪
)⬎␧0/␧1some pa o he wa e will be ansmi ed
h ough he in e ace. This ansmi ed wa e will be an ex-
ao dina y wa e, as i ollows om i s pola iza ion s a e,
and can be exci ed a any equency. This amazing beha io
disappea s when he nonlocal model summa ized in Eq. 共6兲
is used. Indeed, i he nonlocal axial pe mi i i y 共6兲is used
o he wi e medium, we obse e ha no ansmission inside
he wi e medium is possible o k⬍k0.A k⫽k0 ansmis-
sion is possible only in he case o he no mal incidence.
Only i k⬎k0, a e ac ed wa e appea s.
Le us nex conside he guidance o elec omagne ic
wa es in a pa allel-pla e wa eguide in ini e in he xand y
di ec ions and bounded by pa allel pe ec ly conduc ing
planes o hogonal o he zaxis. Sepa a ion be ween he con-
duc ing walls is d. We assume ha his wa eguide is illed
wi h a wi e medium wi h he wi es along he zdi ec ion. We
will conside eigenwa e p opaga ion along he xaxis o he
TM01 mode (Hy,Ex,Ez⫽0). Fo wa eguides illed by a lo-
cal uniaxial dielec ic wi h aniso opy axis along he zdi ec-
ion we ha e om Eq. 共3兲
␧0qx
2⫽␧共k2⫺qz
2兲,qx⫽
冑
␧
␧0
冋
k2⫺
冉
␲
d
冊
2
册
.共13兲
I ␧⬎0, Eq. 共13兲gi es a cu o o k⬍
␲
/dand p opaga ion
o k⬎
␲
/d. In con as , i ␧⬍0, p opaga ion is allowed
when k⬍
␲
/d共and o bidden o k⬎
␲
/d). Wi hin his pass-
band a backwa d wa e (dq/d
␻
⬍0) p opaga es, as one can
see in Fig. 2 共 hin solid lines兲.
This amazing beha io disappea s i one ills he wa e-
guide wi h he analyzed nonlocal wi e medium. Using Eq.
共7兲, we ha e in his case
qx
2⫹qz
2⫽k2⫺k0
2,qx⫽
冑
k2⫺k0
2⫺
冉
␲
d
冊
2
,共14兲
and we ob ain he usual equency beha io : cu o o k
⬍
冑
(
␲
/d)2⫹k0
2and p opaga ion o k⬎
冑
(
␲
/d)2⫹k0
2.An
inc ease o he cu o equency is obse ed compa ed o he
FIG. 2. No malized p opaga ion ac o s in a pa allel-pla e wa e-
guide s no malized equency o di e en ypes o wa eguide
illing: Thin solid lines, uniaxial dielec ic wi h a nega i e pe mi -
i i y; hick solid line, wi e medium; dashed line, emp y wa eguide.
The wi e medium and he uniaxial dielec ic ha e he same k0
⫽
␲
/(2d).
BRIEF REPORTS PHYSICAL REVIEW B 67, 113103 共2003兲
113103-3
case when he e is no illing medium, as one can see in Fig.
2共 hick solid line and dashed line兲.
Le us inally analyze he p opaga ion o plane wa es
along he wi e medium o he pa icula case o qz⫽k.In
his case, he nonlocal pe mi i i y along he zaxis 共6兲be-
comes in ini e. To a oid he singula i y p oblem we use ma-
e ial equa ion o he o m E⫽␧
¯
¯
⫺1D. In his case, he Max-
well equa ions ha e plane-wa e solu ions o all equencies.
Fo hese wa es he ans e se wa e ec o q⬜⫽(qx,qy)Tis
a bi a y. The wa es a e ans e se wi h espec o he wi e
axis: Hz⫽0 and Ez⫽0. The elec ic ield is pa allel o he
ans e se wa e ec o , q⬜⫻E⫽0.
Such wa es can be in e p e ed as ansmission-line modes
p opaga ing along he pa allel wi es. In ac , a se o Nin i-
ni e pa allel wi es can be iewed as a sys em o coupled
ansmission lines. This sys em can suppo Ndegene a e
ansmission-line wa es wi h Hz⫽0, Ez⫽0, and phase con-
s an qz⫽k. The elec ic ield o hese wa es can be ob ained
om
E⫽⫺共ux
⳵
x⫹uy
⳵
y兲
␾
共x,y兲exp共⫺jkz兲,共15兲
whe e
␾
(x,y) is a quasielec os a ic po en ial aking cons an
bu a bi a y alues
␾
n(n⫽1,2,...,N) a each wi e. In ac ,
he plane wa e wi h ans e se wa e numbe q⬜and qz⫽k
co esponds o he ansmission-line wa e wi h
␾
n
⫽
␾
0exp(⫺jq⬜• n), whe e n⫽(xn,yn)Tis he loca ion o
he n h wi e in he ans e se x-yplane.
In summa y, i has been shown ha pa allel wi e media
possess e y s ong spa ial dispe sion e ec s a any equen-
cies, including he e y la ge wa eleng h limi . An analy ical
model o he nonlocal pe mi i i y dyadic o hese media
has been p esen ed and discussed. Inconsis ency o he local
model o pa allel wi e media wi h non anishing wa e-
ec o componen along he wi es has been shown. D ama ic
di e ences in he p edic ed beha io o ha media, a ising
om he use o he con en ional local and/o he nonlocal
model o he pe mi i i y a e shown. Finally, he p oposed
nonlocal model o he pe mi i i y has been ound o be also
sui able o he desc ip ion o he ansmission-line modes o
he s uc u e. We eel ha he epo ed esul s open he ques-
ion o he ole o spa ial dispe sion in he adequa e cha ac-
e iza ion o disc e e me ama e ials as e ec i e media, a
leas i a bi a y di ec ions o p opaga ion and/o pola iza ion
o he elec omagne ic ield should be conside ed in he
analysis. In addi ion, an example has been p esen ed o an
e ec i e medium in which spa ial dispe sion is impo an a
any equency, in con as wi h some commonly assumed
ideas abou he physical ele ance o his e ec .
1L. Landau, E. Li schi z, and L. Pi ae ski, Elec odynamics o
Con inuous Media 共Pe gamon P ess, Ox o d, 1984兲.
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