Applica ion o Huygens’ Me asu aces o he
A bi a y Design o a Leaky-Wa e An enna
Elena Abdo-S´
anchez(1), A iel Eps ein(2), Geo ge V. Ele he iades(3)
[email p o ec ed], [email p o ec ed], [email p o ec ed]
(1)Dp o. Ing. Comunicaciones, E.T.S.I. Telecomunicaci´
on, Uni e sidad de M´
alaga, Andaluc´
ıa Tech, E-29071 M´
alaga, Spain.
(2)And ew and E na Vi e bi Facul y o Elec ical Enginee ing, Technion - Is ael Ins i u e o Technology, Hai a 32000, Is ael.
(3)The Edwa d S. Roge s S . Depa men o Elec ical and Compu e Enginee ing, Uni e si y o To on o, To on o, Canada.
Abs ac —Leaky-wa e an ennas a e guiding s uc u es ha
leak powe along hei leng h. Thei adia ion is mainly cha ac-
e ized by he p opaga ion cons an (leakage ac o and phase
cons an ) o he a eling wa e. In his con ibu ion, a leaky-
wa e an enna based on pa allel-pla e wa eguide is p oposed.
A bi a y con ol o he leakage ac o and he phase con-
s an is achie ed by eplacing he op pla e by an omega- ype
bianiso opic Huygens’ me asu ace, which implemen s he de-
si ed ield ans o ma ion. The heo e ical de i a ion and design
me hodology a e b ie ly desc ibed. Se e al designs wi h di e en
poin ing angles (phase cons an s) and leakage a es ha e been
ca ied ou . Elec omagne ic simula ion esul s alida e he
heo e ical de i a ion, highligh he capabili ies o he s uc u e
and con i m he lexibili y in he design pa ame e s.
I. INTRODUCTION
In he ecen yea s, he e is an inc easing demand on
di ec i e an ennas wi h low p o ile and cos , especially o
applica ions such as au omo i e ada s o sa elli e communi-
ca ions. Good pe o mance is ob ained wi h phased a ays;
howe e , hei eeding ne wo ks a e complica ed, leading o
high cos s. Leaky-wa e an ennas (LWAs), on he con a y,
ha e e y simple eeding, since hey consis on a guiding
s uc u e ha leaks powe while he wa e is being p opaga ed
along i [1]. Plana LWAs ha e ecei ed an inc easing a en-
ion la ely, especially a e he in oduc ion o me ama e ials
and me asu aces, which allowed he enhancemen o hei
cha ac e is ics (such as he mi iga ion o he b oadside e ec )
[2]–[5]. In o de o ob ain a ce ain adia ion pa e n, i is
necessa y o be able o design he p opaga ion cons an o he
leaky mode, γ=β−jα. The phase cons an , β, de e mines
he poin ing angle, whe eas he leakage ac o , α, con ols
he a e o he powe leakage, which se s he ampli ude
dis ibu ion. Independen con ol o hese wo pa ame e s is a
challenge in he design o LWAs.
Con ol on he adia ion pa e n could be achie ed i we
a e able o a bi a ily ans o m he ield inside he guiding
s uc u e in o he desi ed adia ed ield. In his ega d, Huy-
gens’ me asu aces ha e been ecen ly p oposed as a powe ul
ool o a bi a y ield manipula ion [6]–[8]. They consis
o subwa eleng h elec ically- and magne ically-pola izable
pa icles and allow he ul illmen o he equi ed bounda y
condi ions, so ha he desi ed ield ans o ma ion is achie ed.
The e o e, by placing a Huygens’ me asu ace on he op
o he guiding s uc u e, he equi ed bounda y condi ions
o ans o m he guiding mode in o he desi ed leaky mode
can be implemen ed. I has been ecen ly disco e ed ha by
employing omega- ype bianiso opic me asu aces (O-BMSs)
O-BMS
PEC
.
x
zy
d
.E , H
- -
qin
.E , H
+ -
qou
z=-d
z=0
L
k = j
y
+b - a
- -
k = j
y
+b - a
+ +
Fig. 1. T ans e sal sec ion o he p oposed LWA con igu a ion, which
consis s o a pa allel-pla e wa eguide wi h he op pla e being a bianiso opic
me asu ace (O-BMS).
jus one condi ion in he s ipula ion o he ields mus be
me o achie e a bi a y ield ans o ma ion using passi e
and lossless pa icles: local powe conse a ion along he
me asu ace [9]. This is possible due o he addi ional (mag-
ne oelec ic) deg ee o eedom p o ided by he O-BMSs. The
ac ha only one condi ion o a bi a y ield ans o ma ion is
equi ed allows he con ol o he e lec ion and ansmission
coe icien s om he guiding s uc u e o ai h ough he
me asu ace. In his con ibu ion, a pa allel-pla e wa eguide
which inco po a es an O-BMS as he op pla e is p oposed
as a no el LWA wi h ou s anding lexibili y o he design
pa ame e s and sys ema ic design, as highligh ed in [10].
II. THEORY
The explo ed s uc u e (Fig. 1) is a pa allel-pla e wa eg-
uide in which he op pla e is eplaced by an O-BMS (a
z= 0). Fo simplici y, he p oblem conside ed he e is 2D
(∂/∂x = 0). The O-BMS has a leng h in he y-coo dina e o
Land he exci a ion o he esul ing pa allel-pla e wa eguide
is loca ed a y= 0.ds ands o he wa eguide heigh .
A ans e se elec ic (TE) pola ized ield is used as ield
exci a ion (Ey=Ez=Hx= 0). Then, he ans e se ield
componen s abo e (E+
xand H+
y) and below (E−
xand H−
y) he
O-BMS a e ela ed h ough he bianiso opic shee ansi ion
condi ions [11]:
1
2(E+
x+E−
x) = −Zse(H+
y−H−
y)−Kem(E+
x−E−
x)
1
2(H+
y+H−
y) = −Ysm(E+
x−E−
x) + Kem(H+
y−H−
y)
(1)
whe e Zse s ands o he elec ic su ace impedance, Ysm o
he magne ic su ace admi ance and Kem o he magne o-
elec ic coupling coe icien .
The i s s ep o he heo e ical de i a ion o he p oblem
is o ind an exp ession o he elec omagne ic ield inside
he wa eguide ha ul ills Maxwell’s equa ions. Since he
me asu ace design will o ce he bounda y condi ions a
z= 0 o be me , he only es ic ion o he ield below
he O-BMS is o anish a he PEC (z=−d). The e o e, he
ollowing elec ic ield has been s ipula ed:
E−
x=|Ein|(ejk−
z(z+d)−e−jk−
z(z+d))e−jk−
yy(2)
whe e he p opaga ion cons an s a e complex, in o de o le
he s uc u e adia e:
k−
y=β−−jα−;k−
z=β−
z−jα−
z;k−2=k−
y
2+k−
z
2.(3)
The desi ed ield o he egion abo e he me asu ace is
s ipula ed as a leaky mode, simply as
E+
x=|Eou |e−jk+
zze−jk+
yy,(4)
whe e
k+
y=β+−jα+;k+
z=β+
z−jα+
z;k+2=k+
y
2+k+
z
2.(5)
We assume cons an βand αalong y.
I can be demons a ed ha o he powe conse a ion
condi ion o be me (P−
z(y) = P+
z(y)[9]), he ield mus
ha e he same decay along yabo e and below he O-BMS,
i.e. Re[k+
y] = Re[k−
y] = α, whe e αis, indeed, he leakage
ac o . Mo eo e , he cons an |Eou |is gi en by he es o
he pa ame e s [10].
Once we ha e exp essions o s ipula e he ields gi en
desi ed αand β, we can calcula e he me asu ace pa ame e s
{Kem, Ysm, Zse} o achie e he equi ed ield ans o ma-
ion. The only condi ion is ha αmus be he same below
and abo e he me asu ace, bu he es o he pa ame e s
(β+,which de e mines θou ;β−, which de e mines θin; and
he wa eguide heigh d) a e comple ely ee o be se .
I can be shown ha , o cons an α, he me asu ace
pa ame e s {Kem, Ysm, Zse} esul pe iodic, wi h a pe iod
gi en by
p=2π
|β+−β−|.(6)
III. REALIZATION AND SIMULATION RESULTS
To be able o p o e he concep h ough elec omagne ic
simula ion, he me asu ace pa ame e s mus be disc e ized
along y(we ha e used a leng h o λ0/6). To ealize he O-
BMS being compa ible wi h s anda d ab ica ion echniques,
we use asymme ic h ee-laye s ack o impedance shee s [9],
[12]. In his way, we ans o m he local {Kem, Ysm, Zse}
in o he equi ed Z-ma ix o each h ee-laye uni -cell using
(1) and he ela ions be ween he angen ial ields below and
abo e he me asu ace [13]:
E−
x
E+
x=Z11 Z12
Z21 Z22  H−
y
−H+
y.(7)
Then, by applying he ansmission line model o he h ee-
laye s ack o impedance shee s, we ans o m he ma ix [Z]
in o he equi ed alues o he impedance shee s (which esul
o be lossless) {Xbo , Xmid, X op}[9].
The esul ing s uc u e o simula e using he elec omag-
ne ic so wa e HF SS is shown in Fig. 2. The O-BMS
consis s, as can be obse ed, o h ee laye s o eac ance shee s
(which a e simula ed using impedance bounda y condi ions)
Fig. 2. Schema ic o he s uc u e o simula e.
0 2 4 6 8 10
−6
−4
−2
0
2
4
6
y/λ0
Kem
Zse [η]
Ysm [1/η]
Fig. 3. Me asu ace pa ame e s o he Design 1 (wi h a uni -cell leng h o
λ0/6 and a pe iod o 2λ0).
wi h ce ain pe iodici y. To emula e in ini e pla es along he x-
axis, PEC bounda y condi ions a e used on he aces a he zy-
planes. The s uc u e is exci ed and e mina ed by wa epo s.
To illus a e he design p ocedu e and he capabili ies o
he p oposed me hodology, a i s design (Design1) has been
ca ied ou , in which he wa eguide heigh dhas been se o
0.6λ0, he poin ing angle θou has been a bi a ily chosen o
20◦, he pe iod o 2λ0(which de e mines he phase cons an
inside he wa eguide h ough (6) esul ing in θin = 57◦) and
α o 0.02k0. The leng h o he me asu ace has been se o
10λ0in o de o adia e 90% o he powe wi h he chosen
α. Fo hese pa ame e s, he esul ing me asu ace pa ame e s
o implemen a e shown in Fig. 3.
The 2D di ec i i y compa ison be ween heo y and simula-
ion is plo ed in Fig. 4. I can be obse ed ha he s uc u e
adia es a he aimed di ec ion (20◦) and excellen ag eemen
is ound be ween he analy ical p edic ion and he HFSS
esul .
Fig. 5 shows a compa ison o he magni ude o he elec ic
ield. Excellen ag eemen is obse ed no only in he p op-
aga ing mode inside he wa eguide bu also in he adia ing
ield. The exponen ial powe decay along he y−axis, which
co esponds o a cons an α, can be no iced. As can be
obse ed, he p opaga ing mode is g adually leaking powe
−80 −60 −40 −20 0 20 40 60 80
−30
−20
−10
0
10
20
θ [deg]
D [dB]
Analy ical
Simula ion
Fig. 4. 2D di ec i i y o he Design 1 (wi h α= 0.02k0and θou = 20◦).
(a) Analy ical
(b) Simula ion
Fig. 5. Elec ic ield pa e n o he Design 1 (wi h α= 0.02k0and θou =
20◦).
h ough he me asu ace.
In o de o illus a e ha we can se a desi ed α, ano he
design (Design 2) has been ca ied ou by keeping all he
pa ame e s he same as in Design 1 bu educing α o 0.014k0
and inc easing he leng h L= 14λ0 o adia e he same
amoun o powe as in Design 1 bu mo e g adually. Fig.
6 shows he 2D di ec i i y compa ison o his case, in
which he p edic ed inc ease in he di ec i i y and dec ease
in he beamwid h is obse ed wi h espec o Design 1, wi h
excellen ag eemen be ween heo y and simula ion. Fig. 7
shows he compa ison be ween heo y and simula ion o he
magni ude o he elec ic ield o he Design 2. The ield
has p ac ically he same pa e n as o Design 1, since he
pa ame e s ha e been kep he same. The only di e ence is
no iced in he mo e g adual powe leakage, as expec ed.
The LWA can be also designed o adia e in o an a bi a y
angle, by choosing he equi ed β+(β+≈k0sin(θou )).
Then, ei he β−(θin) o he pe iod pcan be a bi a ily se as
well. To illus a e his lexibili y, wo addi ional designs wi h
−80 −60 −40 −20 0 20 40 60 80
−30
−20
−10
0
10
20
θ [deg]
D [dB]
Analy icall
Simula ion
Fig. 6. 2D di ec i i y o he Design 2 (wi h α= 0.014k0and θou = 20◦).
(a) Analy ical
(b) Simula ion
Fig. 7. Elec ic ield pa e n o he Design 2 (wi h α= 0.014k0and
θou = 20◦).
di e en θou ha e been ca ied ou . In bo h cases, θin has
been se o 30◦, so pis di e en in he wo cases, acco ding
o (6). A mo e ex eme angle has been chosen o Design 3,
θou =−50◦, whe eas b oadside adia ion will be illus a ed
wi h Design 4, wi h esul ing pe iods o 0.8λ0and 2λ0,
espec i ely. Due o he low pe iod o Design 3 wi h espec
o he uni -cell leng h o λ0/6, some deg ada ion wi h espec
o he analy ical esul s o his case is expec ed, due o he
sampling o he me asu ace pa ame e s. In bo h designs, he
leng h L, wa eguide heigh d, and leakage ac o αha e been
kep o 10λ0,0.6λ0and 0.02k0, espec i ely.
Fig. 8 shows he 2D di ec i i y compa ison be ween heo y
and simula ion o he Design 3 and Design 4. Some dis-
c epancies in he di ec i i y le els be ween he analy ical and
simula ion esul s a e ound o Design 3 (θou =−50◦),
which a e a ibu ed o he low pe iod (p= 0.8λ0), as
p e iously men ioned. In ac , his pe iod leads o e en less
han i e di e en uni -cells pe pe iod, which migh no be
enough o cap u e he beha io o he con inuous me asu ace.
Howe e , his example was chosen o highligh ha he ield
pa e n inside he me asu ace can be main ained in designs
wi h di e en poin ing angles by using a di e en pe iod (see
Fig. 9). The poin ing angles om he simula ions ag ee wi h
−80 −60 −40 −20 0 20 40 60 80
−30
−20
−10
0
10
20
θ [deg]
D [dB]
Analy ical θou =−50o
Analy ical θou =0o
Simula ion θou =−50o
Simula ion θou =0o
Fig. 8. 2D di ec i i y o Design 3 and Design 4 (wi h θou =−50◦and
θou = 0◦, espec i ely).
(a) θou =−50◦
(b) θou = 0◦
Fig. 9. Simula ed elec ic ield pa e n o he Design 3 and Design 4.
he heo e ical p edic ion. Mo eo e , i is highligh ed ha no
issue wi h b oadside adia ion is ound in his s uc u e, unlike
adi ional LWAs.
IV. CONCLUSIONS
Applica ion o Huygens’ me asu ace o he design o a
LWA has been explo ed. In o de o ha e enough deg ees o
eedom o con ol he leaky mode pa ame e s, omega- ype
bianiso opy mus be in oduced in o he me asu ace. The
heo e ical de i a ion o calcula e he me asu ace pa ame e s
o a bi a y leaky mode wi h cons an leakage ac o has been
shown. Simula ion esul s ha e been ob ained by implemen -
ing he O-BMS using h ee-laye s ack o eac ance shee s.
The lexibili y in he design has been highligh ed h ough
ou designs. I has been shown ha di e en leakage ac o s
can be se while keeping an a bi a y poin ing angle. Mo e-
o e , designs wi h mo e ex eme poin ing di ec ion and e en
b oadside adia ion ha e been illus a ed. The e o e, indepen-
den con ol o he leakage ac o and he poin ing di ec ion
has been success ully achie ed. The design me hodology is
e y powe ul since i allows a sys ema ic design wi h almos
all possible deg ees o eedom (cons an leakage ac o , θin,
θou and e en he wa eguide heigh d).
Fu he esea ch will ocus on ex ending he de i a ion and
design p ocedu e o a modula ed leakage ac o o ha e a
be e con ol on he adia ion pa e n and on ge ing a physical
ealiza ion o he me asu ace wi h expe imen al e i ica ion.
Mo eo e , he scanning capabili ies o he p oposed LWA will
be explo ed.
ACKNOWLEDGMENTS
This p ojec has ecei ed unding om he Eu opean
Union’s Ho izon 2020 esea ch and inno a ion p og amme
unde he Ma ie Sklodowska-Cu ie g an ag eemen No
706334.
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