1
E icien spec al domain MoM o he design
o ci cula ly pola ized e lec a ay an ennas
made o spli ings
Ra ael Flo encio, Ra ael R. Boix, and José Encina ,
Abs ac —The Me hod o Momen s (MoM) in he spec al
domain is used o he analysis o he sca e ing o a plane
wa e by a mul ilaye ed pe iodic s uc u e con aining conduc ing
concen ic spli ings in he uni cell. Basis unc ions accoun ing
o edge singula i ies a e used in he app oxima ion o he
cu en densi y on he spli ings, which makes i possible
a as con e gence o MoM wi h espec o he numbe o
basis unc ions. Since he 2-D Fou ie ans o ms o he basis
unc ions canno be ob ained in closed- o m, judicious icks
(con olled unca ion o in ini e summa ions, in e pola ions, e c.)
a e used o he e icien nume ical de e mina ion o hese Fou ie
ans o ms. The implemen ed spec al domain MoM so wa e
has been used in he design o a ci cula ly pola ized e lec a ay
an enna based on spli ings unde he local pe iodici y condi ion.
The an enna has been analyzed wi h ou spec al domain MoM
so wa e, wi h CST and wi h HFSS, and good ag eemen has
been ound among all se s o esul s. Ou so wa e has p o en
o be a ound 27 imes as e han CST and HFSS.
Index Te ms—Mul ilaye ed media, momen me hods, pe iodic
s uc u es, e lec a ays, ci cula pola iza ion.
I. In oduc ion
REFLECTARRAY an ennas a e an in e es ing al e na i e
o e lec o an ennas and mic os ip a ays owing o
hei e sa ile adia ion pe o mance, low p o ile, ligh weigh ,
ease o ab ica ion, simpli ied eeding sys em, e c. [1]. In he
pa icula case o ci cula pola iza ion (CP) e lec a ays, he
a iable o a ion echnique (VRT) in oduced by Huang and
Pogo zelski is a design app oach ha has p o en o be e y
success ul [2]. In he ame o his echnique, each elemen
o he e lec a ay is o a ed a di e en angle o achie e he
adequa e phase shi in he impinging ci cula ly pola ized wa e
ha makes i possible o gene a e he p esc ibed adia ion
pa e n a e e lec ion. The VRT equi es ha he e lec ion
coe icien s o he wo o hogonal linea componen s o he
impiging wa e elec ic ield a e o equal magni ude and 180◦
ou o phase in o de o keep he sense o ci cula pola iza ion
(LHCP o LHCP o RHCP o RHCP) a e e lec ion [2], [3].
One e lec a ay elemen ha has u ned ou o be especially
sui able o CP applica ions is ha based on spli ings. Han
e al. used he VRT and single spli ings o design a Ka-
band RHCP e lec a ay a 31.75 GHz wi h measu ed c oss-
pola iza ion le el below 40 dB a b oadside [4]. S acked spli
ings o di e en size we e used in [5] o design a dual
equency RHCP e lec a ay ope a ing in he C and Ka bands.
Smi h e al. [6] designed a single laye dual equency CP
e lec a ay by using an elemen wi h wo concen ic spli
ings. The inne ings and he VRT we e used o adjus he
elemen s phase a 29.75 GHz o RHCP adia ion, while he
ou e ings and he VRT we e used a 19.95 GHz o LHCP
adia ion. An ex ension o his wo k was p esen ed in [7]
whe e he dual equency CP e lec a ay was combined wi h
an FSS made o non-spli ings o combine he ope a ion o
he e lec a ay wi h ha o a second CP mic os ip a ay
an enna wo king a L band. Zhao e al. used he single laye
elemen based on wo concen ic ings and he VRT o design
a b oadband CP e lec a ay a a cen e equency o 10 GHz
wi h a bandwid h la ge han 30% o 1-dB gain a ia ion, and
a bandwid h la ge han 40% o an axial a io smalle han
3-dB [8]. In all p e ious pape s dealing wi h CP applica ions,
he elemen s o he e lec a ays consis o symme ic spli
ings wi h wo gaps and one axis o mi o symme y. Zhang
e al. p oposed in [9] he use o non-symme ic single gap spli
ings which can be used as phasing elemen s o bo h linea
pola iza ion (LP) and CP e lec a ay ope a ion. By using his
elemen , a LP e lec a ay was designed a 20 GHz wi h a
bandwid h o 23% o a 1-dB gain a ia ion and an ape u e
e iciency o 60%.
In he design o a CP e lec a ay an enna made o spli
ings, he use o he VRT equi es o adjus he size o he
gaps and hei o ien a ion o e e y single elemen in o de o
achie e he equi ed phase shi in he impinging ci cula ly
pola ized wa e, and in o de o ensu e a phase di e ence
o 180◦in he e lec ion coe icien s o i s wo o hogonal
linea componen s. When es ima ing he gap size and hei
o ien a ion in each elemen , i is cus oma y o assume ha he
elemen s a e loca ed in a pe iodic en i onmen , which is called
he local pe iodici y assump ion [10], [11]. The alidi y o his
assump ion is jus i ied by he ac ha leads o e lec a ay
designs in which he simula ed pe o mance ag ees e y well
wi h he measu ed pe o mance [5], [7], [8]. The design o a
midsize CP e lec a ay an enna made o spli ings unde he
local pe iodici y assump ion equi es he analysis o a la ge
numbe o di e en mul ilaye ed pe iodic s uc u es owing o
he wide a ie y o gap sizes and o ien a ions ha ha e o be
adjus ed in he spli ings o he di e en e lec a ay elemen s
o phase adjus men [7]. Owing o his, a powe ul nume ical
ool is needed o he analysis o hese mul ilaye ed pe iodic
s uc u es.
In his pape we apply he Me hod o Momen s (MoM) in
2
he spec al domain o he analysis o he sca e ing o a plane
wa e by a mul ilaye ed pe iodic s uc u e including concen ic
spli ings in he uni cell. The implemen ed MoM so wa e
is subsequen ly used in he design o a ci cula ly pola ized
e lec a ay an enna made o spli ings. When applying he
spec al domain MoM, he 2-D Fou ie ans o ms o he
basis unc ions a e exp essed as in ini e summa ions o Hankel
ans o ms. This Hankel ans o m app oach was used in he
pas in conjunc ion wi h he spec al domain MoM o he
de e mina ion o he esonan equencies o mic os ip ing
esona o s [12], he inpu impedance and adia ion pa e ns
o single and s acked annula - ing mic os ip an ennas [13],
[14], he esonan equencies o shielded single and coupled
mic os ip ing esona o s [15], [16], and he ada c oss-
sec ion o concen ic annula mic os ip ings [17]. Howe e ,
o he au ho s’ knowledge, i has ne e been used o mi-
c os ip spli ings and o mul ilaye ed s uc u es in pe iodic
en i onmen s. We ha e only ound one pape dealing wi h
he analysis o mic os ip spli ing esona o s in which a
magne ic wall app oxima e model is used o he de e mina ion
o he esonan equencies, and whe e e ec i e dimensions
a e in oduced o accoun o edge e ec s [18]. Whe eas
mos pape s dealing wi h he analysis o mic os ip ing
s uc u es use ei he magne ic wall ca i y mode unc ions
[13], [19] o subsec ional piecewise unc ions [15], [17] as
basis unc ions o he elec ic cu en densi y on he ings,
in his pape we ha e used basis unc ions ha accoun o
edge singula i ies [16], [19] since hese basis unc ions p o ide
a as con e gence o MoM wi h espec o he numbe o
basis unc ions, which implies ha only small MoM ma ices
ha e o be in e ed. Since he Hankel ans o ms o he edge
singula i y basis unc ions canno be ob ained in closed o m,
we ha e used especially ailo ed quad a u e ules o hei
nume ical compu a ion, which p o ide e y accu a e esul s
wi h a small numbe o quad a u e poin s. Also, we ha e
in oduced judicious s op c i e ia o he summa ion o he
in ini e se ies leading o he 2-D Fou ie ans o ms o he
basis unc ions, and we ha e inally ca ied ou Chebyshe
in e pola ions o hese Fou ie ans o ms as a unc ion o he
spec al adial a iable. As a esul o all hese s a egies,
we ha e de eloped a e y e icien spec al domain MoM
app oach o he analysis o mul ilaye ed pe iodic s uc u es
con aining spli ings. The spec al domain MoM so wa e has
been used in he design o a ci cula ly pola ized e lec a ay
an enna and he esul s ob ained ha e been compa ed wi h
esul s p o ided by comme cial so wa es CST and HFSS .
Good ag eemen has been ound be ween he h ee se s o
esul s, ou so wa e being be ween one and wo o de s o
magni ude as e han CST and HFSS .
II. Nume ical p ocedu e
Figs. 1(a) and (b) show a mul ilaye ed pe iodic s uc u e
backed by a g ound plane. Concen ic conduc ing spli ings
a e loca ed in each uni cell, and a e p in ed on he uppe laye .
The cen e o he ings has been chosen o be coinciden wi h
he geome ical cen e o he uni cell in he z=0 plane.
The conduc ing ings and he g ound plane will be assumed
o be pe ec elec ic conduc o s (PEC). In a gene ic case,
he e would be Rconcen ic ings in he uni cell (R=2 in
Fig.1(b)). Also, each ing would be spli in o Sa cs (S=2 in
Fig.1(b)) so ha he o al numbe o a cs in he uni cell would
be L=RS (L=4 in Fig. 1(b)). Le ρ1land ρ2lbe he inne
and ou e adius o he l- h a c espec i ely (l=1,...,L), and
le wl=ρ2l−ρ1lbe he wid h o he l- h a c. The wo ends o
he l- h a c a e cha ac e ized by he angles ϕ1land ϕ2l, such
ha ϕ1l<ϕ2l. The angle ϕkl (k=1,2; l=1,...,L) is he angle
sub ended be ween he k− h end o he l- h a c and he semi-
in ini e line di ec ed along he posi i e xaxis wi h o igin a
he cen e o he ing. The angle ϕkl is aken as posi i e i ,
s a ing om he x-di ec ed semi-in ini e line, i is d awn in
he coun e clockwise sense, and is aken as nega i e i i is
d awn in he clockwise sense (e.g., in Fig. 1(b), ϕ21,ϕ12,ϕ22,
ϕ13 and ϕ23 a e posi i e, bu ϕ11,ϕ14 and ϕ24 a e nega i e).
z
xy
qinc
em
10
em
20
,
,h2
h1
b
bbbbb
(a)
b
a
y
zx
a
j
22
2
23
2
j
13
j
23
j
12
a2
a1
21
j
21
j
11 j
24
j
14
(b)
Fig. 1. Two-laye ed pe iodic s uc u e. The uni cell con ains wo concen ic
spli ings wi h di e en axes o mi o symme y. A plane wa e impinges on
he mul ilaye ed pe iodic s uc u e. (a) Side iew. (b) Top iew.
In he pa icula cases ea ed in his pape , we will assume
ha each indi idual spli ing o he pe iodic s uc u e has an
axis o mi o symme y going h ough he cen e o he ing
(in he case o Fig. 1(b), his means ha ϕ21 −ϕ11=ϕ22 −ϕ12
and ϕ12 −ϕ21=360◦+ϕ11 −ϕ22, and ha ϕ23 −ϕ13=ϕ24
−ϕ14 and 360◦+ϕ14 −ϕ23=ϕ13 −ϕ24). Fo he - h ing
( =1,...,R), his mi o symme y axis makes an angle
α (−180◦≤α <180◦) wi h he semi-in ini e line di ec ed
along he posi i e yaxis wi h o igin a he cen e o he
ing (see Fig. 1(b)). The angle α is aken as posi i e i ,
s a ing om he y-di ec ed semi-in ini e line, i is d awn in he
coun e clockwise sense, and is aken as nega i e i i is d awn
in he clockwise sense (e.g., in Fig. 1(b), α1and α2a e bo h
posi i e). Please no e ha he es ic ion in oduced by hese
mi o symme y axes is no necessa y in he ma hema ical
de i a ions p esen ed in he es o his sec ion.
The concen ic spli ings o he pe iodic s uc u e a e
p in ed on a mul ilaye ed subs a e con aining Nllaye s (Nl=
3
2 in Fig.1(a)). The i- h laye has a hickness hiand a complex
pe mi i i y εi=ε0ε ,i(1 −j an δi) (i=1,...,Nl). In he
ollowing, a ime dependence o he ype ejω will be assumed
and supp essed h oughou .
As commen ed in he in oduc ion, he uni cell o he
mul ilaye ed pe iodic s uc u e o Figs. 1(a) and (b) has been
used as cons i uen elemen o se e al CP e lec a ay an ennas
[6]–[8]. In he design o a CP e lec a ay an enna unde he
local pe iodici y assump ion, each cell is cha ac e ized by
means o a CP 2 ×2 complex e lec ion ma ix, RCP, which
ela es he RHCP and LHCP componen s o he e lec ed and
inciden elec ic ields when a plane wa e impinges on he
cell su ounded by a pe iodic en i onmen . In his pape , his
impinging plane wa e is assumed o p opaga e in an a bi a y
incidence di ec ion gi en by he angula sphe ical coo dina es
θinc and ϕinc (see Fig.1(a)). I E e
RHCP and E e
LHCP a e he RHCP
and LHCP complex componen s o he e lec ed elec ic ield,
and Einc
RHCP and Einc
LHCP a e he RHCP and LHCP complex
componen s o he inciden elec ic ield, hen
E e
RHCP
E e
LHCP !=RCP · Einc
RHCP
Einc
LHCP !(1)
whe e
RCP = RRHCP,RHCP RRHCP,LHCP
RLHCP,RHCP RLHCP,LHCP !.(2)
When w i ing (1), we assume ha he plane wa e impinging
on he pe iodic s uc u e does no gene a e g a ing lobes
a e e lec ion, which equi es ha he condi ion max(a,b)<
λ0/(1 +sin θinc) (λ0=2π/ω √µ0ε0) is ul illed, aand bbeing
he pe iods o he uni cell in he xand ydi ec ions espec i ely
(see Fig. 1(b)).
A. Spec al domain MoM o mula ion
In o de o ob ain he ma ix RCP o (2) o he mul ilaye ed
pe iodic s uc u e o Figs. 1(a) and (b), we need o ob ain he
sca e ed elec ic ield o impinging plane wa es wi h bo h
RHCP and LHCP pola iza ions. These sca e ed elec ic ields
can be ob ained in e ms o cu en densi y exci ed a he
me allized in e ace z=0, J(x,y), by he impinging wa es.
Le Ems(x,y,z) be he elec ic ield gene a ed in all space
by a plane wa e impinging on he mul ilaye ed subs a e o
Figs. 1(a) and (b) in he absence o he conduc ing spli ings.
The cu en densi y J(x,y) induced on he spli ings will be
he solu ion o he ollowing elec ic ield in eg al equa ion
(EFIE) [20]
ˆ
z×"Ems(x,y,z=0) +
+∞
X
m=−∞
+∞
X
n=−∞ZSmn
GE(x−x0,y−y0,z=0,z0=0) ·J(x0,y0)dx0dy0#=0(3)
(x,y)∈S00
whe e Smn (m,n=. . . , −1,0,1, . . .) is he me allized po ion o
he z=0 plane wi hin he mn- h pe iodic uni cell, and GEis
he non–pe iodic dyadic G een’s unc ion o he mul ilaye ed
subs a e [21]. Since J(x,y) is a Floque -pe iodic unc ion o
xand y, in o de o sol e he EFIE o (3), we only need
o de e mine J(x,y) wi hin one uni cell, e. g., he cell C00
co e ing he ec angula domain {0≤x≤a; 0 ≤y≤b}. Fo
ha pu pose, we expand J(x,y) in C00 in e ms o known
basis unc ions Jlq(x,y) (l=1,...,L;q=1,...,Nb) as shown
below
J(x,y)=
L
X
l=1
Nb
X
q=1
clqJlq(x,y) (4)
whe e Jlq(x,y) (q=1,...,Nb) is he se o Nbbasis unc ions
used o app oxima e he cu en densi y on he l- h a c o he
C00 uni cell. When (4) is in oduced in (3) and Gale kin’s
e sion o MoM is applied, he ollowing sys em o linea
equa ions is ob ained o he unknown coe icien s clq [20]
L
X
l=1
Nb
X
q=1
Γkl,pqclq =ekp (k=1,...,L;p=1,...,Nb).(5)
I we in oke Pa se al’s iden i y o 2-D Fou ie ans o ms,
he MoM ma ix en ies Γkl,pq o (5) can be exp essed in he
spec al domain as double in ini e summa ions gi en by [22]
Γkl,pq =ab
+∞
X
m=−∞
+∞
X
n=−∞he
Jd
kp(kxm,kyn)∗i
·e
G
E,c
(kx=kxm,ky=kyn,z=0,z0=0)
·e
Jd
lq(kxm,kyn) (6)
whe e e
G
E,c
(kx,ky,z=0,z0=0) is he con inuous 2-D
Fou ie ans o m o GE(x,y,z=0,z0=0) [21], kxm =
k0sin θinc cos ϕinc +2πm/a,kyn =k0sin θinc sin ϕinc +2πn/b,
k0=2π/λ0and e
Jd
lq(kxm,kyn) is he disc e e 2-D Fou ie
ans o m o Jlq(x,y), which is gi en by
e
Jd
lq(kxm,kyn)=1
ab ZS00
Jlq(x,y) e−j(kxm x+kyny)dxdy (7)
Finally, he coe icien s ekp o he sys em o equa ions (5)
can be ob ained in he spec al domain as
ekp =−ab he
Jd
kp(kx0,ky0)∗i ·Ems(x,y,z=0)
×e−jk0(sin θinc cosϕinc x+sin θinc sinϕincy)(8)
whe e he ac o e−jk0(sin θinc cosϕinc x+sin θinc sinϕincy)has been explic-
i ly included in (8) o abso b he dependence o Ems(x,y,z=0)
on xand y.
Equa ions (5) o (8) p o ide he spec al domain MoM
o mula ion o he de e mina ion o he ec o unc ion J(x,y)
o (3) and (4). The p oblem wi h his o mula ion is ha he
disc e e 2-D Fou ie ans o ms e
Jd
lq(kxm,kyn) o (6) (de ined in
(7)) canno be ob ained in closed o m, e en o he simples
choice o basis unc ions in (4) (i.e., a cons an alue o he
wo componen s o Jlq(x,y) on he su ace o he spli ings
o Figs. 1(a) and (b)). In he ollowing subsec ion, we will see
how o ackle his p oblem.
B. Basis unc ions and 2-D Fou ie ans o ms
Fig. 2 shows he l- h a c (l=1,...,L=4) o he uni cell
C00 o Fig. 1(b). In Fig. 2 we ha e de ined a shi ed sys em
o coo dina es {x0,y0,z0}wi h o igin a he cen e o he uni
4
b
a
y
z
x
1
l
z’
y’
x’
2l
a/2
b/2
j
1l
j
2l
Fig. 2. Top iew o he l- h a c in he le uni cell o Fig. 1(b). Radial and
angula dimensions o he l- h a c a e shown.
cell in he plane z=0. We a e going o in oduce shi ed
pola coo dina es ρ0and ϕ0, which a e ela ed wi h he o iginal
coo dina es xand yas
x=a
2+ρ0cos ϕ0(9)
y=b
2+ρ0sin ϕ0(10)
The uni ec o s linked o he shi ed pola coo dina es a e
gi en by
ˆρ0=cos ϕ0ˆ
x+sin ϕ0ˆ
y(11)
ˆϕ0=−sin ϕ0ˆ
x+cos ϕ0ˆ
y(12)
In he spli ings used in CP e lec a ay an ennas, he
condi ion wl=ρ2l−ρ1l(ρ1l+ρ2l)/2 (see Fig. 2) is usually
ul illed (i.e., he wid h o he ings is usually much smalle
han hei mean adius) [6]–[8]. Assuming his condi ion holds,
we a e going o neglec he adial componen o he basis
unc ions Jlq(x,y) o (4), Jlq(x,y)·ˆρ0, by compa ison wi h hei
azimu hal componen , Jlq(x,y)·ˆϕ0. Also, we will assume ha
he azimu hal componen o Jlq(x,y) can be ac o ed in e ms
o independen unc ions o ρ0and ϕ0, i.e., we will assume
ha he unc ions Jlq(x,y) can all be ma hema ically w i en
as
Jlq(x,y)= l(ρ0)glq(ϕ0)ˆϕ0(13)
(ρ1l<ρ0<ρ2l;ϕ1l<ϕ0<ϕ2l)
As we will see in Sec ion III, he assump ion shown in (13)
is jus i ied by he ac ha leads o alues o he ma ix RCP in
(2) ha ma ch he alues p o ided by he comme cial so wa e
CST . Al hough he adial componen , Jlq(x,y)·ˆρ0, o Jlq(x,y)
has been neglec ed in (13), we would like o poin ou ha his
componen can be easily accommoda ed in he ma hema ical
o mula ion p esen ed in he es o his subsec ion.
Since he unc ions glq(ϕ0)ˆϕ0o (13) a e pe iodic unc ions
o ϕ0wi h pe iod 2π, hese unc ions can be expanded as
Fou ie se ies o ϕ0as shown below
glq(ϕ0)ˆϕ0=
+∞
X
i=−∞e
gi
lq ejiϕ0(ϕ1l<ϕ0<ϕ2l) (14)
whe e
e
gi
lq =1
2πZϕ2l
ϕ1l
glq(ϕ0)ˆϕ0e−jiϕ0dϕ0(15)
Now, le us in oduce he pola disc e e spec al coo dina es
kρ,mn and kϕ,mn gi en by
kρ,mn =q(kxm)2+(kyn)2(16)
kϕ,mn =a c an kyn
kxm !(17)
I (9) o (14), (16) and (17) a e in oduced in (7), a e some
manipula ions, i is possible o exp ess e
Jd
lq(kxm,kyn) as
e
Jd
lq(kxm,kyn)=2πe−jkxma+kynb
2
ab
×(e
g0
lq ˜
0
l(kρ=kρ,mn)+
+∞
X
i=1
e−jiπ/2˜
i
l(kρ=kρ,mn)
ejikϕ,mne
gi
lq +e−jikϕ,mn e
gi
lq∗)(18)
whe e ˜
i
l(kρ) (i=0,1, . . .) a e Hankel ans o ms o o de io
he unc ions l(ρ0) o (13), which can be exp essed as
˜
i
l(kρ)=Zρ2l
ρ1l
Ji(kρρ0)ρ0 l(ρ0)dρ0,(19)
and whe e he unc ion Ji(·) o (19) is a Bessel unc ions o
i s kind and o de i.
Al hough he exp ession (18) o he compu a ion o
e
Jd
lq(kxm,kyn) looks simple han (7), i p esen s wo d awbacks.
Fi s , he unc ions ˜
i
l(kρ) canno be ob ained in closed o m,
e en o he simples choice o l(ρ0) (e.g., a cons an alue
in he in e al ρ1l<ρ0<ρ2l). Second, (18) in ol es he de e mi-
na ion o an in ini e se ies, which has a dele e ious impac on
he compu a ion o e
Jd
lq(kxm,kyn) by means o (18).
In his pape , he unc ions chosen o Jlq(x,y) a e en i e
domain basis unc ions which accoun o he singula i ies o
J(x,y) a he edges o he a cs o Fig. 1(b). I is well known
ha hese unc ions ensu e a as con e gence o he spec al
domain MoM wi h espec o he numbe o basis unc ions
(LNbin (4)) [16], [19], [20], as will be demons a ed in Sec ion
III. The pa icula unc ions chosen o l(ρ0) and glq(ϕ0) can
be w i en as
l(ρ0)=1
1−2
ρ2l−ρ1lhρ0−ρ2l+ρ1l
2i2
(20)
(l=1,...,L)
glq(ϕ0)=s1− 2
ϕ2l−ϕ1lϕ0−ϕ2l+ϕ1l
2!2
×Uq−1 2
ϕ2l−ϕ1lϕ0−ϕ2l+ϕ1l
2!(21)
(l=1,...,L;q=1,...,Nb)
whe e Uq−1(·) is a Chebyshe polynomial o second kind and
deg ee q−1.
The in eg als o (15) can be ob ained in closed o m o
he unc ions glq(ϕ0) o (21), and hei exp ession is gi en
5
in he Appendix A. Howe e , he in eg als o (19) canno
be ob ained in closed o m o he unc ions l(ρ0) o (20).
Fo una ely, we ha e checked hese la e in eg als can be
nume ically ob ained wi hin e y easonable CPU imes by
means o Gauss-Chebyshe quad a u e ules. In pa icula ,
when he unc ions l(ρ0) o (20) a e in oduced in (19), he
in eg als can be ew i en as
˜
i
l(kρ)=Z+1
−1
hi
l(kρ, )
√1− 2d (22)
whe e
hi
l(kρ, )=ρ2l−ρ1l
2Jikρρ2l−ρ1l
2 +ρ2l+ρ1l
2
×ρ2l−ρ1l
2 +ρ2l+ρ1l
2 (23)
The in eg als o (22) a e amenable o be compu ed by means
o Gauss-Chebyshe quad a u e ules [23, Eqn. 25.4.38]. In
pa icula , he in eg als should be app oxima ely ob ained by
means o he closed- o m o mula
˜
i
l(kρ)≈
Nqp
X
j=1
ηjhi
l(kρ, = j) (24)
whe e ηj=π/Nqp and j=cos (2 j−1)π/2Nqp(j=
1,...,Nqp) ( j=1,...,Nqp). Since he unc ions hi
l(kρ, ) a e
non-singula smoo h unc ions o in he in e al −1≤ ≤+1,
a low numbe o quad a u e poin s, Nqp, should su ice o
ob ain ˜
i
l(kρ) wi h a la ge accu acy. This has been con i med
by nume ical simula ions, as will be shown in Sec ion III.
Conce ning he in ini e se ies o (18), i s con e gence is
s ongly dependen on he alue o kρ,mn. The la ge he
alue o kρ,mn, he la ge he numbe o e ms ha has o be
e ained in he se ies o (18) o an accu a e de e mina ion o
e
Jd
lq(kxm,kyn). Le us see how he con e gence o his in ini e
se ies depends on kρ,mn. I we assume ha a≈b(which holds
in p ac ical cases o mos e lec a ay an ennas), by i ue o
he mean alue heo em o de ini e in eg als, we can w i e
ha (see [23, Eqn. 9.3.1])
|˜
i
l(kρ=kρ,mn)||i| ∝Jikρ,mn ρ2l+ρ1l
2|i|
≈Ji kρ,mna
2!|i|∝ ekρ,mna
4i!i
=eiln(ekρ,mna/4i)(25)
which indica es ha he se ies o (18) has an exponen-
ial con e gence p o ided ekρ,mna/4i≤1, i. e., p o ided
i≥ekρ,mna/4. Nume ical simula ions ha e shown ha he
in ini e se ies o (18) has o be added in he in e al 1 ≤
i≤ekρ,mna/4+8 o an accu a e es ima ion o e
Jd
lq(kxm,kyn).
The e o e, he la ge kρ,mn, he la ge he numbe o e ms o
be e ained in he se ies.
In p ac ice, he compu a ion o Γkl,pq equi es o unca e
he in ini e summa ions o (6) wi hin he in e als −Nmax ⩽
m,n⩽+Nmax. I we assume ha a≈b, he maximum alue
o kρ,mn we will ha e o use in (6) a e he unca ion will be
(see (16))
kmax
ρ,mn =q(kmax
xm )2+(kmax
yn )2≈√22πNmax
a(26)
which means he maximum numbe o e ms ha will be
necessa y o e ain in he in ini e se ies o (18) will be
imax =ekmax
ρ,mna
4+8≈πeNmax
√2
+8 (27)
Fo he cases ea ed in he esul s sec ion, we ha e used
Nmax =50, which leads o imax ≈310 when using (27). Fo
ou pu poses, his is he wo s case scena io in he e alua ion
o (18), i.e., any e alua ion o e
Jd
lq(kxm,kyn) wi hin −Nmax ⩽
m,n⩽+Nmax will equi e o e ain a mos 310 e ms in he
compu a ion o he in ini e se ies o (18) when Nmax =50.
Apa om he analy ical s op c i e ium es ablished abo e
o he summa ion o he se ies o (18) (1 ≤i≤ekρ,mna/4+8),
one addi ional ac ha helps o educe he CPU ime equi ed
in he compu a ion o his se ies is a well known p ope y o
Bessel unc ions, acco ding o which Ji(x)≈0 i x<x h =C hi
when i>N h. Nume ical simula ions show his p ope y holds
o N h a ound 10 and C h a ound 1. Acco ding o (19), his
p ope y o Bessel unc ions implies ha ˜
i
l(kρ=kρ,mn)≈0
in (18) when i>N h and (kρ,mna)/2<C hi, which makes i
unnecessa y he e alua ion o a la ge numbe o summa ion
e ms in he nume ical compu a ion o he se ies leading o
e
Jd
lq(kxm,kyn) by means o (18). In Sec ion III we will p esen
a nume ical me hod ha makes i possible o easily es ima e
wo new quan i ies d h(l) and b h(l) (which a e a unc ion o
ρ1land ρ2l) o which ˜
i
l(kρ)≈0 i kρa<d h(l)(i−b h(l))
and i≥b h(l). Acco ding o his, i will only be necessa y o
compu e ˜
i
l(kρ=kρ,mn) in he in e al
kmin
ρ,mn(i)=d h(l)(i−b h(l))
a≤kρ≤kmax
ρ,mn (28)
when i≥b h(l). Fo i<b h(l), we will ha e o compu e ˜
i
l(kρ=
kρ,mn) in he in e al kmin
ρ,mn(i)=0≤kρ≤kmax
ρ,mn.
The in eg als ˜
i
l(kρ) o (19) a e a unc ion o kρ,ρ1land
ρ2l o each alue o i. In mos applica ions o e lec a ay
an ennas he dis ances ρ1land ρ2l(l=1,...,L) emain ixed
om elemen o elemen and he angles ϕkl (k=1,...,S;l=
1,...,L) and α ( =1,...,R) a e a ied o ob ain he equi ed
phase shi s. So, a la ge amoun o CPU ime can be sa ed in
he design o a la ge e lec a ay an enna i he unc ions ˜
i
l(kρ)
(l=1,...,L;i=0,...,imax) a e in e pola ed as a unc ion
o kρ o ixed alues o ρ1land ρ2l, and he in e pola ions
a e subsequen ly used when adjus ing he angles ϕkl and α
in each elemen . The in e pola ions ha e o be pe o med
in he in e al kmin
ρ,mn(i)≤kρ≤kmax
ρ,mn o each alue o l
and i. In p ac ice, we ha e di ided he equi ed in e pola ion
in e als in o Nin subin e als, and we ha e used Chebyshe
polynomials up o se en h-deg ee in he in e pola ion in each
subin e al. In Sec ion III we will see ha a maximum numbe
o Nin =80 subin e als su ices o ob ain in e pola ed alues
o ˜
i
l(kρ) wi hin ou signi ican igu es, and ha he numbe
o equi ed subin e als Nin dec eases as iinc eases, which is
a consequence o he inc ease o kmin
ρ,mn(i) (and he e o e, o he
educ ion o he wid h o he o iginal in e pola ion in e al) as
iinc eases.
III. Nume ical esul s and alida ions
Figs. 3(a) and (b) show he ela i e e o s made in he e al-
ua ion o ˜
0
l(kρ) and ˜
100
l(kρ) by means o (24) as a unc ion

6
(a)
(b)
Fig. 3. Magni ude o he ela i e e o s in he compu a ion o (a) ˜
0
l(kρ) and
(b) ˜
100
l(kρ) by means o (24). The ela i e e o s a e plo ed as a unc ion o
he numbe o quad a u e poin s Nqp employed in (24). Pa ame e s: a=b=5
mm; ρ1l=1.85 mm; ρ2l=2.05 mm.
o kρ, and as a unc ion o he numbe o Gauss-Chebyshe
quad a u e poin s Nqp employed in (24). The e e ence alues
used o ˜
0
l(kρ) and ˜
100
l(kρ) ha e been compu ed by means
o he double exponen ial quad a u e ule wi h 203 in eg a ion
poin s (le el o he quad a u e ule M=5), since his is a
nume ical p ocedu e which is pa icula ly accu a e when used
in he in eg a ion o unc ions wi h in eg able singula i ies a
he end poin s o he in eg a ion in e al [24]. Please no e ha a
alue o Nqp equal o 10 su ices o p o ide 4 signi ican igu es
in he alues o ˜
0
l(kρ) and ˜
100
l(kρ) in nea ly all cases, which
indica es ha (24) p o ides alues ha a e accu a e enough
o ˜
i
l(kρ) wi hin a e y low CPU ime consump ion. In he
es o he esul s o his Sec ion, we will always use Nqp =10
in (24).
In Figs. 4(a) and (b) we plo he h eshold alues k h
ρ(i) as
a unc ion o i, whe e k h
ρ(i) a e he alues o kρ o which
|˜
i
l(kρ)/˜
0
l(kρ)|<10−4i kρ<k h
ρ(i) and |˜
i
l(kρ)/˜
0
l(kρ)|>10−4
i kρ>k h
ρ(i). The e o e, o e e y alue o i, he h eshold
alue k h
ρ(i) indica es he alue o kρbelow which ˜
i
l(kρ) can
be conside ed o be anishing. Please no e he plo o k h
ρ(i)
as a unc ion o i i s e y well o a s aigh line. So we ha e
ca ied ou a linea leas squa es i ing o k h
ρ(i) as a unc ion
o igi en by
k h
ρ(i)a≈d h(l)(i−b h(l)) (29)
which should be alid o i≥b h(l). Fo he leas squa es
i ing, we ha e used Nsam samples o iin ascending o de ,
ik(k=1,...,Nsam) he i s sample being i1=20. In
i
0 20 40 60 80 100 120 140 160 180 200
kρa
0
100
200
300
400
500
kρ
h(i)a
(a)
i
0 20 40 60 80 100 120 140
kρa
0
100
200
300
400
500
kρ
h(i)a
(b)
Fig. 4. The solid line k h
ρ(i)as ands o he h eshold alues o kρa o
which ˜
i
l(kρ) is negligible i kρa<k h
ρ(i)a, and ˜
i
l(kρ) is non-negligible i
kρa>k h
ρ(i)a.k h
ρ(i)ais plo ed as a unc ion o i. Pa ame e s: a=b=5 mm;
ρ1l=1.85 mm and ρ2l=2.05 mm in (a); ρ1l=1.20 mm and ρ2l=1.40 mm in
(b).
he case o Fig. 4(a), i u ns ou ha d h(l)≈2.343 and
b h(l)≈9.484 wi h a coe icien o de e mina ion 2=0.9999.
In he case o Fig. 4(b), i u ns ou ha d h(l)≈3.388 and
b h(l)≈8.371 wi h a coe icien o de e mina ion 2=0.9998.
These esul s indica e ha he alues o d h(l) and b h(l) a e
sligh ly dependen on he alues o he inne and ou e adii o
he ings, ρ1land ρ2l, which a e he endpoin s o he in eg a ion
in e al o (19). Since he i ing o k h
ρ(i) o a s aigh line o
i≥b h(l) is e y good, in p ac ice a small alue o he numbe
o samples o k h
ρ(i), Nsam, su ices o he de e mina ion o
d h(l) and b h(l). Once hese alues o d h(l) and b h(l) a e
known, one can use (28) o es ima e he in e al o alues o
kρin which he unc ions ˜
i
l(kρ) p esen non-negligible alues
wi h a iew o compu ing e
Jd
lq(kxm,kyn) ia (18).
Figs. 5(a) and (b) show he e o s made in he in e pola ion
o ˜
0
l(kρ) and ˜
100
l(kρ) in he in e al kmin
ρ,mn(i)≤kρ≤kmax
ρ,mn
when his in e al is di ided in o Nin subin e als o equal
wid h, and Chebyshe polynomials up o se en h-deg ee a e
subsequen ly used o ca y ou he in e pola ion in each subin-
e al. Please no e ha whe eas 80 subin e als a e equi ed o
he de e mina ion o ˜
0
l(kρ) wi h an accu acy o 4 signi ican
igu es in he whole in e al kmin
ρ,mn(i)≤kρ≤kmax
ρ,mn, only 40
subin e als a e equi ed o he de e mina ion o ˜
100
l(kρ)
wi h he same accu acy. The explana ion o his is ha
whe eas he whole in e pola ion in e al o ˜
0
l(kρ) is oughly
0≤kρa.450, he in e pola ion in e al o ˜
100
l(kρ) is much
smalle , 200 .kρa.450, which is a consequence o he ac
ha ˜
100
l(kρ) is negligible o kρa.200 as shown in Fig.4(a).
Fig. 6(b) shows he phase cu es equi ed o he design
7
(a)
(b)
Fig. 5. Magni ude o he ela i e e o s in he in e pola ion o (a) ˜
0
l(kρ)
and (b) ˜
100
l(kρ) as a unc ion o he numbe o subin e als Nin used in he
in e pola ion. Pa ame e s: a=b=5 mm; ρ1l=1.85 mm; ρ2l=2.05 mm.
o a CP e lec a ay an enna (see (2)) made o single spli
ings (see Fig.6(a)) a a equency o 19.95 GHz. By i ue o
he VRT o [2], a linea phase a ia ion is ob ained o bo h
∠RRHCP,RHCP and ∠RLHCP,LHCP as a unc ion o α2, which is
he o a ion angle o he mi o symme y axis o he spli
ing. In o de o keep he sense o CP a e e lec ion in
a e lec a ay an enna, we need a phase di e ence o 180◦
be ween he e lec ion coe icien o he componen o he
elec ic ield along he mi o symme y axis o he spli ing
and he e lec ion coe icien o i s o hogonal componen [2],
[3]. To ensu e his phase di e ence is main ained, he leng h o
he a cs in he e lec a ay elemen has o be sligh ly adjus ed
as α2is a ied [6]. This adjus men o he leng h o he a cs
as a unc ion o α2is shown in Fig. 6(c). In Fig. 6(b) we
show he con e gence o ∠RRHCP,RHCP and ∠RLHCP,LHCP as a
unc ion o he numbe o basis unc ions Nbused in he wo
a cs o he spli ings. I can be seen ha jus h ee basis
unc ions pe a c su ice o achie e con e gence (in ac , he
esul s ob ained wi h he h ee basis unc ions co esponding
o q=1,2,3 in (21) o e lap hose ob ained when Nb=7 and
1≤q≤7), which means ha only 6×6 MoM ma ices ha e o
be in e ed when analyzing he mul ilaye ed pe iodic s uc u e
con aining single spli ings in he uni cell. In o de o alida e
he esul s ob ained wi h he spec al MoM so wa e desc ibed
in Sec ion II, hese esul s a e compa ed wi h esul s p o ided
by he comme cial so wa e CST . Excellen ag eemen is
ound be ween bo h se s o esul s.
In Fig. 7(b) we plo he phase cu es o a dual- equency
CP e lec a ay an enna made o concen ic spli ings (see
b
a
y
z
x
y’
x’
23
j
23
j
13
a
2
13
(a)
α2 (deg)
20 40 60 80 100 120 140 160 180
RRHCP,RHCP and RLHCP,LHCP (deg)
-300
-200
-100
0
100
200
300
RLHCP,LHCP q=1,2
RLHCP,LHCP q=1,2,3
RLHCP,LHCP q=1,...,7
RLHCP,LHCP CST
RRHCP,RHCP q=1,2
RRHCP,RHCP q=1,2,3
RRHCP,RHCP q=1,...,7
RRHCP,RHCP CST
(b)
α2 (deg)
20 40 60 80 100 120 140 160 180
ϕ23 −ϕ13(deg)
147
147.5
148
148.5
149
149.5
150
150.5
(c)
Fig. 6. (a) Single spli ing. (b) Phase o RRHCP,RHCP and RLHCP,LHCP o
he e lec a ay elemen o (a) as a unc ion o he o a ion angle o he ing
α2. Con e gence is s udied wi h espec o he numbe o basis unc ions
(q=1,...,7) used in he wo a cs o he spli ings. Ou esul s ob ained wi h
he spec al domain MoM a e compa ed wi h esul s ob ained wi h CST . (c)
Va ia ion o he leng h o he a cs o he spli ing wi h he o a ion angle
α2 o ensu e a phase di e ence o 180◦in he e lec ion coe icien s o he
wo o hogonal linea componen s o he CP impinging wa es. Pa ame e s:
a=b=5 mm; ρ13=1.85 mm; ρ23=2.05 mm; h1=0.787 mm, ε ,1=2.2,
an δ1=0.0009; h2=0 mm; θinc =30◦,ϕinc=0◦; =19.95 GHz.
Fig. 7(a)). While he ou e ings a e in ended o adjus he
phase a 19.95 GHz, he inne ings a e in ended o adjus he
phase a 29.75 GHz as in [6]. In Fig.7(b) we plo he cu es o
phase adjus men a 29.75 GHz. Since his phase adjus men
is con olled by he inne ings, in he applica ion o he VRT
∠RRHCP,RHCP and ∠RLHCP,LHCP a e a ied by o a ing he mi o
symme y angle o he inne spli ing, α1, while keeping
ixed he mi o symme y axis o he ou e ing α2. No e
ha whe eas ∠RRHCP,RHCP and ∠RLHCP,LHCP a e linea unc ions
o α2in Fig. 6(b), ∠RRHCP,RHCP and ∠RLHCP,LHCP a e no any
longe linea unc ions o α1in Fig. 7(b). This is due o he
ac ha whe eas in Fig.6(b) he whole e lec a ay elemen is
8
b
a
y
z
x
13
y’
x’
j
13
j
21
j
11
21
11
a1
a
2
j
23
23
(a)
α1 (deg)
20 40 60 80 100 120 140 160 180
RRHCP,RHCP and RLHCP,LHCP (deg)
-300
-200
-100
0
100
200
RLHCP,LHCP q=1,2
RLHCP,LHCP q=1,2,3
RLHCP,LHCP q=1,...,7
RLHCP,LHCP CST
RRHCP,RHCP q=1,2
RRHCP,RHCP q=1,2,3
RRHCP,RHCP q=1,...,7
RRHCP,RHCP CST
(b)
α1 (deg)
20 40 60 80 100 120 140 160 180
ϕ21 −ϕ11(deg)
145
150
155
160
165
(c)
Fig. 7. (a) Two concen ic spli ings. (b) Phase o RRHCP,RHCP and
RLHCP,LHCP o he e lec a ay elemen o (a) as a unc ion o he o a ion
angle o he inne ing α1. Con e gence is s udied wi h espec o he numbe
o basis unc ions (q=1,...,7) used in he ou a cs o he wo spli ings.
Ou esul s ob ained wi h he spec al domain MoM a e compa ed wi h esul s
ob ained wi h CST . (c) Va ia ion o he leng h o he a cs o he inne
ing wi h he o a ion angle α1 o ensu e a phase di e ence o 180◦in he
e lec ion coe icien s o he wo o hogonal linea componen s o he CP
impinging wa es. Pa ame e s: a=b=5 mm; ρ13=1.85 mm; ρ23=2.05 mm;
ϕ13=−75.2◦mm; ϕ23=75.2◦mm; α2=0◦;ρ11=1.20 mm; ρ21=1.40 mm;
h1=0.787 mm, ε ,1=2.2, an δ1=0.0009; h2=0 mm; θinc =30◦,ϕinc=0◦;
=29.75 GHz.
o a ed o phase adjus men , in Fig. 7(b) only he inne pa
o he elemen is o a ed o phase adjus men , and he e o e
he p opo ionali y be ween phase and o a ion angle p o en in
[2] is los . Fo una ely, he o a ion o α1in Fig. 7(b) p o ides
enough phase ange o e lec a ay design. In Fig.7(c) we plo
he adjus men s in he leng h o he inne a cs ha a e needed
o keep a phase di e ence o 180◦be ween he e lec ion
coe icien s o he wo o hogonal componen s o he elec ic
ield o he impinging CP wa es. These adjus men s a e la ge
han in he case o Fig. 7(b), which is a ibu ed o he ac
TABLE I
CPU imes (seconds) equi ed o he gene a ion o 31 poin s in he cu es
o Figs.6(b)and 7(b). The CPU imes a e o ou MoMso wa e and CST .
Figu e TMoM
CPU TCST
CPU
Fig. 6(b) 89.8 2383
Fig. 7(b) 152.9 2486
TABLE II
CPU imes (seconds)con ibu ions (compu a ion o Hankel ans o ms,
in e pola ion o Hankel ans o ms,calcula ion o 2-D Fou ie ans o ms
o he basis unc ions and e alua ion o ma ix en ies) o he o al CPU
ime equi ed by ou MoMso wa e o he gene a ion o one poin in
Figs.6(b)and 7(b).
Figu e THT TINT T2DFT TME TMoM
TOTAL
Fig. 6(b) 0.398 0.0005 2.868 0.007 3.290
Fig. 7(b) 0.570 0.0008 4.871 0.020 5.477
ha he ou e a cs o he elemen a e no modi ied du ing
he p ocess o phase adjus men . As in Fig. 6(b), in Fig. 7(b)
con e gence o he spec al domain MoM so wa e is achie ed
wi h jus h ee basis unc ions pe a c (which equi es he
in e sion o 12×12 MoM ma ices), and excellen ag eemen is
ound be ween he spec al domain MoM so wa e and CST .
We ha e compu ed he CPU ime equi ed by CST o
gene a e 31 poin s o he cu es o Figs. 6(b) and 7(b), TCST
CPU,
and he CPU ime equi ed by he spec al domain MoM
so wa e when 3 basis unc ions pe a c a e employed, TMoM
CPU .
When unning he spec al MoM so wa e, we ha e used he
in e pola ed e sions o ˜
i
l(kρ), and we ha e included in TMoM
CPU
he CPU ime equi ed o compu e k h
ρ(i) and in e pola e he
unc ions ˜
i
l(kρ). The esul s ob ained o TCST
CPU and TMoM
CPU
a e shown in Table I. These CPU imes ha e been ob ained
in a lap op compu e wi h p ocesso In el Co e i7-6700HQ
a 2.6 GHz wi h ou co es and 32 GB o RAM memo y.
The MoM code has been w i en in FORTRAN language.
Acco ding o he esul s o Table I, he spec al MoM so wa e
is ypically 27 imes as e han CST in he analysis o he
pe iodic s uc u es s udied in Fig. 6(b), and a ound 16 imes
as e han CST in he analysis o he pe iodic s uc u es
o Fig. 7(b). When he spec al domain MoM desc ibed in
Sec ion II is applied o he analysis o one single pe iodic
s uc u e con aining spli ings, he e a e di e en nume ical
s eps ha ing di e en CPU ime con ibu ions o he o al CPU
ime equi ed by MoM, TMoM
TOTAL. These CPU ime con ibu ions
include he CPU ime equi ed o compu e he Hankel ans-
o ms ˜
i
l(kρ) ha a e needed o he in e pola ions by means o
(24), THT, he CPU ime equi ed o he in e pola ions o he
Hankel ans o ms by means o Chebyshe polynomials, TINT,
he CPU ime equi ed o he e alua ion o he 2-D Fou ie
ans o ms o he basis unc ions by means o (18), T2DFT,
and he CPU ime equi ed o he de e mina ion o he MoM
9
ma ix en ies by means o (6), TME. Table II shows hese
di e en CPU ime con ibu ions o TMoM
TOTAL ( he CPU imes
ha e been ob ained wi h he same compu e used o Table
I). In pa icula , he CPU imes p esen ed in Table II a e o
he pe iodic s uc u e analyzed in Fig. 6(b) when α2=92.9◦
(ϕ23 −ϕ13=150.4◦in Fig. 6(c)), and o he pe iodic s uc u e
analyzed in Fig. 7(b) when α1=92.9◦(ϕ21 −ϕ11=150.0◦
in Fig. 7(c)). No e ha he mos impo an con ibu ion o
he o al MoM CPU ime is T2DFT, which is a ound 88% o
TMoM
TOTAL. This jus i ies he e o s ca ied ou in Sec ion II o
compu e he Hankel ans o ms ˜
i
l(kρ) appea ing in (18) in
an e icien way. I we had no op imized he compu a ion o
˜
i
l(kρ), he me hod p esen ed in his pape would no ha e been
compe i i e by compa ison wi h comme cial so wa e. Once
he unc ions e
Jd
lq(kxm,kyn) o (18) ha e all been compu ed, he
CPU imes equi ed o compu e he double summa ions o (6)
and o sol e he sys em o equa ions o (5) a e negligible (no e
ha he maximum size o he MoM ma ices o be in e ed is
12 ×12). The nex impo an con ibu ion o he o al MoM
CPU ime is THT, which ep esen s a ound 11% o TMoM
TOTAL.
Table II shows he con ibu ion o he ime equi ed o he
in e pola ion o he Hankel ans o ms o TMoM
TOTAL is negligible.
A his poin , we should emembe ha he compu a ion and
in e pola ion o he Hankel an o ms has o be ca ied ou only
once in he design o a e lec a ay an enna whe e he inne
and ou e adii o he ings ρ1land ρ2l(l=1,...,L) emain
cons an in all he elemen s o he an enna. The e o e, we can
neglec he con ibu ion o THT o he CPU ime equi ed in
he design o a whole e lec a ay an enna. In ou s udy o
he compu a ional pe o mance o he spec al domain MoM
so wa e as a unc ion o he di e en inpu a iables in ol ed,
we ha e inally analyzed he compu a ional complexi y o
TMoM
TOTAL as a unc ion o he numbe o quad a u e poin s used
in (24), Nqp, as a unc ion o he numbe o e ms e ained in
in ini e se ies o (18), imax, and as a unc ion o he numbe
o basis unc ions used in he applica ion o MoM, Nb. We
ha e ound ha he e ec o Nqp on TMoM
TOTAL is i ele an o
3≤Nqp ≤15, ha he spec al MoM is oughly o complexi y
O((imax)2) o 48 ≤imax ≤200, and ha he spec al MoM is
oughly o complexi y O(Nb) o 1 ≤Nb≤7.
Based on he cu es o Figs. 6(b), 6(c), 7(b), and 7(c), a
dual- equency pencil beam CP e lec a ay an enna has been
designed. The an enna adia es LHCP a 19.95 GHz and RHCP
a 29.75 GHz as in he case o he an enna designed in [6].
The designed an enna is ci cula and consis s o 5024 elemen s
a anged in a 80 ×80 g id wi h cell size 5 mm ×5 mm ( he
diame e o he an enna is 400 mm). The an enna is in ended o
p oduce a ocused beam in he di ec ion θb=30◦and ϕb=0◦
(see [25, Fig. 5.a]). I is illumina ed by a co uga ed ci cula
eed-ho n wi h i s phase cen e loca ed a he coo dina es
x=−150 mm, y=0 mm, z=259.8 mm wi h espec o a
coo dina e sys em wi h o igin a he cen e o he e lec a ay
(see [25, Fig. 5.a]). The ho n is assumed o adia e LHCP
wa es a 19.95 GHz and RHCP wa es a 29.75 GHz. The
adia ion pa e n o he ho n is modelled as a unc ion cos7(θ),
which p o ides an illumina ion le el a he e lec a ay edges
12 dB below he maximum. The an enna elemen s a e dual
concen ic spli ings as hose shown in Fig.7(a). In he design
Azimu h angle (deg)
-50 -40 -30 -20 -10 0 10 20 30 40 50
Gain (dB)
-20
-10
0
10
20
30
40
Copola LHCP q=1,2,3
Copola LHCP CST
Copola LHCP HFSS
C osspola LHCP q=1,2,3
C osspola LHCP CST
C osspola LHCP HFSS
(a)
Ele a ion angle (deg)
-50 -40 -30 -20 -10 0 10 20 30 40 50
Gain (dB)
-20
-10
0
10
20
30
40
Copola LHCP q=1,2,3
Copola LHCP CST
Copola LHCP HFSS
C osspola LHCP q=1,2,3
C osspola LHCP CST
C osspola LHCP HFSS
(b)
Fig. 8. Radia ion pa e ns in (a) he azimu h plane and (b) he ele a ion plane
o a dual- equency CP e lec a ay based on an elemen wi h wo concen ic
spli ings. Plo s a e p esen ed o he analysis o he e lec a ay unde he
local pe iodici y assump ion wi h bo h ou spec al domain MoM so wa e
(solid and dash-do ed lines), CST (dashed and do ed lines) and HFSS (×
and +). The esul s plo ed a e o LHCP adia ion a 19.95 GHz.
o he an enna a 19.95 GHz, he dimensions and o ien a ions
o he ou e ings ha e been adjus ed in acco dance wi h
Figs. 6(b) and 6(c), while assuming he inne ings we e no
p esen . Howe e , when he an enna has been designed a
29.75 GHz, bo h he ou e and he inne ings ha e been aken
in o accoun . The dimensions and o ien a ions o he ou e
ings ha e been ixed a he alues ob ained o he design
a 19.95 GHz, and he dimensions and o ien a ions o he
inne ings ha e been adjus ed in acco dance wi h Figs. 7(b)
and 7(c). The design o he an enna has been ca ied ou by
means o he spec al domain MoM so wa e desc ibed in
Sec ion II unde he local pe iodici y condi ion, while using he
exac incidence angle in each e lec a ay elemen . Once he
e lec a ay an enna has been designed, i has been analyzed
wi h he spec al domain MoM so wa e (using h ee basis
unc ions pe a c wi h q=1,2,3 in (21)), and wi h he
wo comme cial so wa es CST and HFSS , unde he local
pe iodici y assump ion. In he analysis, he exac incidence
angle has been conside ed in each e lec a ay elemen . Figs.8
and 9 show he adia ion pa e ns ob ained in he azimu h and
ele a ion planes a 19.95 GHz and 29.75 GHz espec i ely.
The esul ing gain is 35.88 dBi a 19.95 GHz and 39.42 dBi
a 29.75 GHz. And he an enna e iciency is o 56% a 19.95
GHz and 57% a 29.75 GHz. In Fig. 8 he ag eemen be ween
ou MoM esul s, CST and HFSS is excellen . In Fig. 9
he ag eemen is excellen o he copola componen o he