P og ess In Elec omagne ics Resea ch, Vol. 141, 149–160, 2013
AN ALTERNATIVE MODEL FOR WAVE PROPAGATION
IN ANISOTROPIC IMPEDANCE-MATCHED METAMA-
TERIALS
Consuelo Bell e -Ceb e os*and Ma celo Rod iguez-Dan a
Depa amen o de Fisica Aplicada III, Escuela Tecnica Supe io de
Ingenie ia, Uni e sidad de Se illa, Camino de los Descub imien os s/n,
Se ille 41092, Spain
Abs ac —The p opaga ion o ligh in an aniso opic impedance-
ma ched me ama e ial is s udied in he ame o geome ical op ics.
We p o e ha di ec ions o ields ¯
D,¯
Band ¯ ( ay eloci y) a e a iad
o conjuga e di ec ions wi h espec o he in e se ela i e dielec ic
pe mi i i y enso and cons i u es a local basis, whose ecip ocal one
is o med by di ec ions o ¯
E,¯
H ields and wa e- ec o ¯
k. Consequen ly,
bo h dual bases a e in insically ela ed o he physical p ope ies o
medium. We ha e iden i ied hese bases wi h di ec and ecip ocal
bases o a cu ilinea coo dina es sys em, showing ha physics de ines
geome y. This iden i ica ion p o ides a powe ul ool o sol e wo
kinds o p oblems (di ec and in e se ones) ha cu en ly a ise: In
di ec p oblems, medium p ope ies a e gi en and i su ices o know
˜ε= ˜µ enso a e e y poin , o ob ain he wa e s uc u e. In
in e se p oblems, medium p ope ies mus be ound o he ays o
p opaga e along p esc ibed ajec o ies. The p ocedu e is applied o
an illus a ing example.
1. INTRODUCTION
When dealing wi h laws o p opaga ion o monoch oma ic plane
wa es, he so called “op ical ans o ma ion heo y” in s a ing an
equi alence p inciple be ween elec omagne ic pa ame e s o a physical
medium exp essed in Ca esian coo dina es and hei analogues in
acuum ob ained om a cu ilinea coo dina e ans o ma ion, pu s in
co espondence he Euclidean inhomogeneous and aniso opic physical
Recei ed 5 June 2013, Accep ed 6 July 2013, Scheduled 12 July 2013
* Co esponding au ho : Consuelo Bell e -Ceb e os ([email p o ec ed]).
150 Bell e -Ceb e os and Rod iguez-Dan a
space, wi h ano he homogeneous, iso opic and ma e ee space, bu
endowed wi h a locally la me ic. Thus, he ay ajec o y in he
physical medium is associa ed wi h a geodesic cu e in i ual space.
E e y analogy in Physics is a guide ha enable us o p edic
hidden ela ions and ha monies. Thus, i we conside ha s eam lines
in i o a ional luid mo ion a e he analogues o ays in geome ical
op ics, he in isibili y p oblem in op ics becomes simila o ha o
he lux o a luid s eam in he p esence o an obs acle. Analogy
and p inciple o equi alence allow, in la p oblems, o eco e he old
heo y o complex po en ial (con o mal mapping) and, consequen ly,
ma hema ical analysis is conside ably simpli ied [1].
Ne e heless,“geome ical physics” in hese media can be
unde s ood, in ou opinion, no only as a di ec consequence (“all
occu s as i ”) o he in a iance o Maxwell equa ions in coo dina e
ans o ma ions [2–4], bu any hing else: physical p ope ies o a
medium in insically de ine he mos adequa e geome y.
In gene al, a isible equencies (430–790 THz), na u al media
exhibi a homogeneous and iso opic magne ic beha iou oge he
wi h an inhomogeneous and aniso opic dielec ic one. Ce ain
me ama e ials ( o which heo ies de eloped and desc ibed in his
pape can be applied) when subjec ed o ex e nal ac ions, obey linea
cons i u i e equa ions: ¯
D=ε0˜ε¯
E;¯
B=µ0˜µ¯
H, wi h equal ela i e
pe mi i i y and pe meabili y enso s, (˜ε= ˜µ). The speci ici y o
beha iou laws enable us o endow he space wi h a me ic biuni ocally
linked o he me ama e ial physics. These p ope ies can no be
applied o na u al media wi h dielec ic aniso opy and magne ic
iso opy [5].
In he p esen al e na i e ea men , i has been p o ed ha in
any poin o hese me ama e ials (in absence o cha ges and cu en s)
he wa e on (o he ay) is ge ing dis o ed in such a way ha
¯
D,¯
B ields and ay eloci y ¯ (o ¯
E,¯
Hand ¯
k) cons i u e a local
basis o conjuga ed di ec ions wi h espec o ela i e pe mi i i y o
pe meabili y enso e alua ed a ha poin .
2. ALTERNATIVE ANALYTICAL MODEL
In his sec ion, om ela i e pe mi i i y (o pe meabili y) enso
and om laws o e olu ion in he medium, we a e in ended o
es ablish ec o bases consis ing o conjuga ed di ec ions. These bases,
in insically ela ed o physical p ope ies, a e pu in co espondence
wi h local ec o bases ha , in cu ilinea coo dina es, a e going o
de ine he medium geome y.
P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 151
2.1. Conjuga e Di ec ions
Le ˜
Ibe a symme ic enso o second o de , wi h componen s Iij
e e ed o a Ca esian e e ence ame {¯
i1,¯
i2,¯
i3}and le ˆνand ˆµbe
any wo di ec ions. The bound ec o o ˆνby he enso [6] is ec o
¯
Γνde ined as ¯
Γν=˜
I·ˆν.
The in insic componen , σνo ec o ¯
Γνis equal o i s p ojec ion
on o ec o ˆνin such a way ha σν=¯
Γν·ˆν. The p ojec ion o ec o
¯
Γνon o ˆµis ¯
Γν·ˆµand as a consequence o Cauchy heo em [7] we
a i e o: ¯
Γν·ˆµ=¯
Γµ·ˆν. Two di ec ions ˆνand ˆµa e named conjuga e
i [8, 9]: ¯
Γν·ˆµ=¯
Γµ·ˆν= 0 (1)
Figu e 1 p o ides a simple geome ical in e p e a ion: A ele an
p ope y o Cauchy quad ic (de ined by σν=cons ) [7] is ha i s
no mal a e e y poin is collinea wi h bound ec o ¯
Γν o any
di ec ion ˆν. The o hogonali y o ec o s ¯
Γνand ˆµimplies he ela ion
o conjuga ion.
O
x
P
H
Q
π
π1
2µ
Γν
z
y
ν
Figu e 1. Cauchy’s quad ic associa ed wi h a second-o de enso ˜
I.
The o hogonali y o ec o s ¯
Γν=˜
I·ˆνand ˆµimplies ha uni ec o s
ˆνand ˆµa e conjuga e di ec ions. Vec o ˆµlies in plane π2pa allel o
he angen plane o he ellipsoid a “P”.
I mus be no ed ha p incipal di ec ions o a enso a e also
conjuga e di ec ions. Conjuga ion a ises as a b oade concep han
ha o p incipal di ec ions (a pa icula case, whe e ¯
Γνis collinea wi h
ˆν), bu keeping he algeb aic p ope y o diagonalisa ion o enso s, i
exp essed in co a ian componen s.
152 Bell e -Ceb e os and Rod iguez-Dan a
2.2. Laws o E olu ion
The double Fou ie ans o m ¯
(ω, ¯
k) o a ec o ield ¯
F( , ¯ ), unc ion
o bo h ime and posi ion is:
¯
¡ω, ¯
k¢=Zd d3¯ ¯
F( , ¯ )ei(ω −¯
k·¯ )(2)
and i can be shown ha [10]:
∂¯
F
∂ → −iω (ω, ¯
k); ∇· ¯
F( , ¯ )→i¯
k· (ω, ¯
k) (3)
∇× ¯
F( , ¯ )→i¯
k× (ω, ¯
k) (4)
Applying hese exp essions o Maxwell equa ions o sou ce- ee media,
we ha e:
¯
kׯ
E=ω¯
B;¯
kׯ
H=−ω¯
D(5)
¯
k·¯
D= 0; ¯
k·¯
B= 0 (6)
Bea ing in mind ha wa e ec o ¯
k(wa e- on g adien ) and ay
eloci y ¯ (collinea wi h Poyn ing ec o ), a e gi en by [11]:
¯
k=(¯
Dׯ
B)ω
W; ¯ =(¯
Eׯ
H)
W(7)
(whe e Wis he elec omagne ic ene gy densi y), om Maxwell
equa ions [12] o om Exp ession (7), he ollowing equa ions in ¯ -
domain a ise:
¯ ׯ
B=−¯
E; ¯ ׯ
D=¯
H(8)
¯ ·¯
E= 0; ¯ ·¯
H= 0 (9)
In o de o comple e he abo e equa ions, we add some speci ic
cons i u i e ela ions o a i icial media (me ama e ials), whe e ela i e
pe mi i i y enso (˜εij) and ela i e pe meabili y one (˜µij) a e
iden ical when e e ed o he same ec o basis; consequen ly he
iden i y holds also o in e se enso s ˜ε−1
ij = ˜µ−1
ij = ˜τij (in wha ollows,
˜τ). In o he wo ds,
¯
D=ε0˜ε·¯
E;¯
E=˜τ·¯
D
ε0
;¯
B=µ0˜ε·¯
H;¯
H=˜τ·¯
B
µ0
(10)
I is immedia e o s a e ha in hese media, elec omagne ic ene gy
densi y is gi en by W=¯
E·¯
D=¯
B·¯
H; i.e., elec ic ene gy densi y
coincides wi h magne ic one, and consequen ly, ¯
k·¯ =ω.
Unde he assump ions o p opaga ion o monoch oma ic (ω
= cons an ) plane wa es in he absence o cu en densi ies and
cha ges, Equa ions (5), (6), (8), (9) and (10) enable us o de i e some
di e en geome ical s uc u es and hei co esponding physical laws
o e olu ion.
P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 153
2.2.1. Geome ical S uc u e o Fields
a) In o de o desc ibe he laws o e olu ion o ields, we choose a local
base (P, ¯ei/{i= 1,2,3}), consis ing o ec o s ¯ei, collinea wi h ¯
D,¯
B
ields and ¯ ( ay eloci y), espec i ely. We will p o e ha his basis
is conjuga ed wi h espec o he enso ˜τ. The chosen ec o s a e:
¯e1=¯
D
|¯
D|; ¯e2=¯
B
|¯
B|; ¯e3=¯
|¯ |; wi h ¯ei=hiˆui;h1=h2=h3=1 (11)
and √g= (¯e1ׯe2)·¯e3. Equa ions (8) and (9) p o e ha ¯
E(¯
H) is
o hogonal o ¯
B(¯
D) and ¯ , and cons i u i e laws (10) p o e ha ¯
Eis
collinea wi h ¯
Γu1= ˜τ·ˆu1(and ¯
His collinea wi h ¯
Γu2= ˜τ·ˆu2). Thus,
ε0¯
E
|¯
D|= ˜τ·ˆu1=¯
Γu1;µ0¯
H
|¯
B|= ˜τ·ˆu2=¯
Γu2(12)
F om he abo e men ioned o hogonali y ela ions, Equa ion (12)
and Cauchy heo em, we can deduce ha ¯
Γu1·ˆu2=¯
Γu1·ˆu3=¯
Γu2·ˆu1=
¯
Γu3·ˆu1=¯
Γu2·ˆu3=¯
Γu3·ˆu2= 0. In o he wo ds, he elec ic
displacemen ield ¯
D, he magne ic induc ion ield ¯
Band ay eloci y
¯ locally p opaga e along conjuga ed di ec ions o he in e se ela i e
pe mi i i y enso o he medium. F om he abo e analysis, i yields
he choice o he ecip ocal local base (P, ¯ej/{j= 1,2,3}), ha consis s
o ec o s collinea wi h he bound ec o s o di ec ions ˆu1, ˆu2, ˆu3and
de ined as:
¯ej=
¯
Γuj
hjτj
=˜τ·¯ej
h2
jτj−→ i yields ¯ei·¯ej=δj
i(13)
wi h τj=¯
Γuj·ˆuj(τjcoincide wi h he eigen alues o he enso , when
ec o s ˆuia e o hogonal).
In hese ames o e e ence (bases ¯eio ¯ei), ields ha e only a
componen (co a ian o con a a ian ). Thus,
¯
D=D1¯e1;¯
B=B2¯e2; ¯ = 3¯e3;¯
E=E1¯e1;¯
H=H2¯e2; (14)
wi h D1=|¯
D|/h1;B2=|¯
B|/h2; 3=|¯ |/h3;E1=|¯
D|h1τ1/ε0;
H2=|¯
B|h2τ2/µ0, and he in e se ela i e pe mi i i y enso (in
co a ian componen s) is w i en as:
˜τ=τij ¯ei⊗¯ejwi h τij =h2
iτjδj
i(15)
Wi h espec o he physical meaning o τ1and τ2, we e ain
he exp essions o elec ic and magne ic ene gy densi ies in a linea
medium [11, 13]:
We=¯
E·¯
D
2;Wm=¯
H·¯
B
2(16)

154 Bell e -Ceb e os and Rod iguez-Dan a
Since τj=¯
Γuj·ˆuj, we can w i e:
τ1=¯
Γu1·ˆu1=ε0¯
E·¯
D
|¯
D|2=We
W0
e
;τ2=¯
Γu2·ˆu2=µ0¯
H·¯
B
|¯
B|2=Wm
W0
m
(17)
Coe icien τ1is he a io be ween he elec ic ene gy densi y in
he medium, We, and he elec ic ene gy densi y, W0
e, ca ied by he
wa e i p opaga ion we e in acuum and ob iously, i coincides wi h
he in e se ela i e pe mi i i y associa ed wi h he ˆ
u1di ec ion. The
same applies o τ2, bu e e ed o magne ic ene gy densi y.
In o de o know he physical meaning o e e y ec o o he
basis ha desc ibe co a ian componen s o ields, i emains only o
in es iga e wha ec o ¯e3 ep esen s. F om (7), (13), (14) and aking
in o accoun he linea cha ac e o ope a o ˜τ, we can w i e,
˜τ·(¯
Eׯ
H)=E1H2˜τ¡¯e1ׯe2¢=E1H2²123˜τ·¯e3=E1H2
√gh2
3τ3²312¯e1ׯe2
=h2
1h2
2h2
3τ1τ2τ3
ε0µ0gD1B2(¯e1ׯe2)= h2
1h2
2h2
3τ1τ2τ3
ε0µ0g¡¯
Dׯ
B¢(18)
whe e componen s ²123 =²312 o Le i-Ci i a enso a e equal o
1/√g. F om Equa ions (7), (13), (18), and aking in o accoun ha
ε0µ0= 1/c2, we can w i e:
˜τ·ˆ =τ1τ2τ3h2
1h2
2h2
3c2
gω ¯
k=⇒¯e3=τ1τ2h2
1h2
2c2
gω|¯ |¯
k(19)
We ha e shown ha ec o ¯e3has he di ec ion o he wa e- ec o
¯
k, and hen ¯
k=k3¯e3. To summa ize,
¯
D=D1¯e1;¯
B=B2¯e2; ¯ = 3¯e3;¯
E=E1¯e1;¯
H=H2¯e2;¯
k=k3¯e3(20)
Then, ec o s o he ecip ocal base ¯eilie along ¯
E,¯
Hand ¯
k
di ec ions. The e is an in insic geome ical s uc u e, associa ed wi h
p opaga ion o plane wa es in hese media in such a way ha he
Euclidean space is endowed, a e e y poin P, wi h wo dual bases
de ined om physical p ope ies o he medium. Table 1 and Figu e 2
show his geome ical s uc u e.
Table 1. Di ec base ec o s and ecip ocal base ec o s ob ained
om enso ˜τ.
Di ec basis Recip ocal basis
¯e1collinea wi h ¯
D¯e1colllinea wi h ¯
E
¯e2collinea wi h ¯
B¯e2collinea wi h ¯
H
¯e3collinea wi h ¯ ¯e3collinea wi h ¯
k
P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 155
k
B
ED
H
e1e1
e2
e3
e2
e3
P
Figu e 2. Local ecip ocal bases ¯eiand ¯eiassocia ed wi h he
s uc u e o a ligh plane wa e p opaga ing h ough an aniso opic
impedance-ma ched medium.
b) I espec i e o any hing gi en abo e, we may assume ha he
space is desc ibed by a sys em o cu ilinea coo dina es, q1,q2,q3
ela ed o Ca esian ones, x1,x2,x3 h ough he equa ions:
qj=qj(x1, x2, x3)⇔xj=xj(q1, q2, q3) wi h j= 1,2,3,(21)
ha allow us o de ine, i he ans o ma ion is admissible [7], wo
ecip ocal local bases a any poin Pas:
Di ec basis: (P, ¯ej)/¯ej=∂¯
∂qj
=
k=3
X
k=1
∂xk
∂qj
¯
ikwi h j={1,2,3}(22)
Dual basis: (P, ¯ej)/¯ej=∇qj=
k=3
X
k=1
∂qj
∂xk
¯
ikwi h j={1,2,3}(23)
whe e ecip oci y ela ion ¯ei·¯ej=δj
iholds. T ans o ma ion ules
be ween base ec o s a e gi en by ¯ei=gij ¯ej, ¯ei=gij ¯ej, whe e
gij = ¯ei·¯ejis he me ic enso .
This geome y, de ined by me ic enso gij, sugges s o iden i y
local bases (11) and (13), in insically ela ed o he physical p ope ies
o medium, wi h (22) and (23) o he jus desc ibed a bi a y
cu ilinea coo dina es sys em. F om his co espondence, physics
de ines geome y.
Assuming he equali y o local bases, we ha e a powe ul ool o
sol e wo kinds o p oblems ha cu en ly a ise: di ec and in e se
156 Bell e -Ceb e os and Rod iguez-Dan a
ones. Fo di ec p oblems, medium p ope ies a e gi en and i su ices
o know ˜ε= ˜µ enso a e e y poin , because he me hod p o ides
he wa e s uc u e. In in e se p oblems, we mus ind he medium
p ope ies o he ays o p opaga e along p esc ibed ajec o ies. I ay
ajec o y is gi en, i.e., geome y is known, we can de e mine medium
p ope ies ˜τ(See Sec ion 3: Applica ion).
2.2.2. Ray E olu ion and Fe ma P inciple
Unlike na u al op ical aniso opic media (wi h dielec ic aniso opy
and magne ic iso opy), in hese media wi h ˜ε= ˜µ, he e is an only
mode o p opaga ion and an only ay eloci y ¯ , o e e y wa e- ec o
¯
k.
F om Equa ion (19) and since in his case, ¯
k·¯ =ω, we ob ain,
¯ ·˜τ·¯ =τ1τ2τ3h2
1h2
2h2
3c2
g(24)
In acco dance wi h he de ini ion o ay index o e ac ion n [11] as
n =c/ , and emembe ing ha ¯ is angen o he ay ajec o y:
¯ = d¯ /dl = ¯
, we can w i e:
n2
=c2
2=g
h2
1h2
2h2
3τ1τ2τ3
(ˆu3·˜τ·ˆu3) (25)
We ha e jus ound he exp ession o he ay index as:
n =√g
h1h2h3µ¯
·˜τ
τ1τ2τ3·¯
¶1/2
(26)
wi h ˆu3≡¯
=d¯
dl wi h |d¯ |=dl. Since n is a unc ion bo h o
posi ion ¯ and o he di ec ion o ec o d¯ /dl, Equa ion (26) shows
he complexi y o ligh p opaga ion in inhomogeneous and aniso opic
media, e en o he ma e ials in discussion, o which he e is an
only mode o p opaga ion, and we can sol e o n . Unlike he
iso opic case, he dis ance be ween wo in ini ely nea poin s depend
on hei posi ion and also on hei ela i e o ien a ion. Consequen ly,
equa ions o e olu ion can be ound as geodesics in a Finsle space [14],
in which he line elemen is gi en by ds =n (¯ , d¯ /dl)dl. When
alues o n a e subs i u ed in o he Fe ma ‘s a ia ional p inciple in
o de o use Hamil onian and Lag angian o malism, a p oblem a ises
because con en ional Legend e ans o m, would ob ain an iden ically
anishing Hamil onian om Lag angian, which is homogeneous o
deg ee one in d¯ /dl. Ne e heless, Fe ma p inciple is in a ian unde
epa ame e ising, hen pa ame e “a c leng h” lmay be subs i u ed
by op ical pa h, in such a way ha dl =n dσ [15], and a non-singula
P og ess In Elec omagne ics Resea ch, Vol. 141, 2013 157
Lag angian a ises as L=n2
(¯ , d¯ /dσ). Consequen ly, equa ions o
e olu ion can be ound as ex emals o he unc ional,
δZn2
dσ = 0 (27)
whe e dis ance is measu ed by a el ime (op ical pa h).
3. APPLICATION
Le us illus a e he abo e heo y wi h an applica ion. Since di ec
p oblems ha e been mo e discussed, we deal wi h an in e se one:
Which op ical p ope ies mus he medium ha e o ligh - ays o
p opaga e along a p esc ibed ajec o y? Fo ins ance, we wan o
cha ac e ize medium p ope ies in such a way ha e e y plane wa e
ha p opaga es along an axis, say z(wa e ec o ¯
kpa allel o zaxis),
yields a ay desc ibing an helicoidal ajec o y o axis z. Fo ha
pu pose, we in oduce he ollowing coo dina es sys em (See Figu e 3).
q1=z
θ;q2=%;q3=z(28)
exp essed in cylind ical coo dina es: %,θ, z. We see ha coo dina e
plane q3= cons an desc ibes he wa e- on and coo dina e line q3is
along he desi ed ay.
Vec o s o he di ec basis a e:
¯e1=∇q1=−z
%θ2ˆuθ+1
θˆuz; ¯e2=∇q2= ˆu%; ¯e3=∇q3= ˆuz(29)
2
2
2
3
e
line q
line q
q= cons .
q= cons .
3
Figu e 3. Coo dina e su aces and coo dina e lines o cu ilinea
coo dina es sys em (q1, q2, q3), de ined as q1=z/θ;q2=%;q3=z,
whe e cylind ical coo dina es %,θand za e used. In his case,
coo dina e plane q3= cons an desc ibes he wa e- on and coo dina e
line q3is along he desi ed ay.