Post on 01-Aug-2020
Research ArticleElectromagnetic Mathematical Modeling of3D Supershaped Dielectric Lens Antennas
L Mescia1 P Bia2 D Caratelli34 M A Chiapperino1 O Stukach4 and J Gielis5
1Dipartimento di Ingegneria Elettrica e dellrsquoInformazione Politecnico di Bari Via Edoardo Orabona 4 70125 Bari Italy2EmTeSys SRL 70122 Bari Italy3The Antenna Company Nederland BV High Tech Campus 5656 AE Eindhoven Netherlands4Institute of Cybernetics Tomsk Polytechnic University 843 Sovetskaya Street Tomsk 634050 Russia5University of Antwerp Groenenborgerlaan 171 2020 Antwerp Belgium
Correspondence should be addressed to L Mescia lucianomesciapolibait
Received 1 October 2015 Revised 14 January 2016 Accepted 7 February 2016
Academic Editor Thierry Floquet
Copyright copy 2016 L Mescia et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The electromagnetic analysis of a special class of 3D dielectric lens antennas is described in detail This new class of lens antennashas a geometrical shape defined by the three-dimensional extension of Gielisrsquo formula The analytical description of the lens shapeallows the development of a dedicated semianalytical hybrid modeling approach based on geometrical tube tracing and physicaloptic In order to increase the accuracy of the model the multiple reflections occurring within the lens are also taken into account
1 Introduction
During the last years lens antennas have attracted theattention of various researchers and companies because oftheir potential use in several application fields such as highfrequency wireless communication systems [1 2] millimeterwave imaging [3] space [4] smart antennas [5] radio-astronomy [6] and radar systems [1] In particular in[1] a flat-top radiation pattern at 60GHz is achieved byconsidering a 2D arbitrary shape lens antenna close in twometal half disks useful for high speedWi-Fi 80211ad devicesAnother application of such class of antennas is illustratedin [3] where the authors presented a lens-coupled patchantenna array for imaging applications This architectureconsists of a two-dimensional array featuring a monolithicmicrowave integrated circuit (MMIC) front-end In [1] theauthors presented a fully polarimetric integrated monopulsereceiver working in W-band frequency range The proposedstructure is composed by 2 times 2 array of slot-ring antennascovered by a hemispherical dielectric lens
Lens antennas can be easily integrated in electronic cir-cuits thanks to the relevant mechanical and thermal stabilitycombined with the possibility of shaping the radiated beam
by changing the lens geometry During the last years manyresearch activities have been devoted to the developmentof 3D dielectric lenses characterized by canonical or axial-symmetrical geometry [7 8] and only a few scientific stud-ies have been dedicated to the analysis of more complexgeometries [9ndash11] In particular in [10] the authors presenta compact lens antenna with arbitrary shape working inmillimeter wave frequency band the design being carriedout while neglecting the internal reflections occurring withinthe lens However in order to enhance the accuracy of thetheoretical model based on geometrical optic (GO) it isessential to consider the effects of the internal reflectionsespecially when high values of electric permittivity have tobe used To this end in [12 13] an accurate mathematicalmodel is illustrated implementing the effects of the second-order internal reflections for canonical lens geometry
The aim of this work is to present in detail a semiana-lytical model to design a new class of shaped dielectric lensantennas whose geometry is described by Gielisrsquo superfor-mula [14 15] The developed mathematical model is basedon the tube tracing approximation [16] The electromagneticfield inside the lens is evaluated by using the GO approxi-mation and considering the effect of the internal reflections
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8130160 10 pageshttpdxdoiorg10115520168130160
2 Mathematical Problems in Engineering
P
PFF
rd
x
rFF
r0
z
ki
120579i
120579t
y
kt
n
Figure 1 Geometrical structure of the electromagnetic systembased on dielectric lens antenna
In this way an improvement of the numerical accuracy canbe obtained Moreover the evaluation of the electromagneticfield radiated from the lens has been carried out applying thephysical optics (PO) approximation In particular accordingto the equivalence principle the radiation outgoing the lensis evaluated by the radiation of the equivalent electric andmagnetic currents on the lens surface The current distribu-tion is calculated by making use of the Fresnel transmissioncoefficient on the lens surface in accordance with the GOprocedure
Thanks to Gielisrsquo formulation it is possible to generate awide range of 3D shapes in a simple and analytical way bychanging a reduced number of parameters The use of thisformulation allows also an easy description of the geometri-cal characteristic of the antenna in terms of lens volume areacurvature radius and aspect ratio Moreover the analyticalrepresentation of the lens shape provides a number of benefitspertaining to the evaluation of the main physical quantitiesinvolved in the electromagnetic propagation equations Inthis way a reduced computation effort is required and moreaccurate results can be obtained
2 Mathematical Model
The geometrical structure considered in the proposed elec-tromagneticmodel consists of a dielectric lens antenna placedon a perfectly electric conductor (PEC) circular plate havingradius 119903119889 (see Figure 1)
The metal plate acts as a ground plane and mechan-ical support of the radiating structure while reducing thebackscattered radiation level The lens is illuminated by theelectromagnetic field emitted by a primary radio source suchas open ended waveguide patch antenna horn antennaor a coaxial probe integrated in the metal plate The tubetracing approach based on the GO approximation is used tomodel the electromagnetic propagation inside the homoge-neous dielectric lensThis approximation allows a significantsimplification of the mathematical model since it makes itpossible to reduce the solution of Maxwellrsquos equations toa geometrical problem As a result the adoption of suchmethod allows the simulation of electrically large structureswith a lower computational effort in comparison to full-wavenumerical methods such as Finite-Difference-Time-Domain(FDTD) and Finite-Element-Method (FEM)
z
y
x
Pmminus1A
Pmminus1B
Pmminus1CAmminus1
PmA
PmB
Am PmC
Figure 2 Area changes from119860119898minus1 to119860119898 of the tube surface duringthe propagation inside the lens
The applicability condition of such method requires thatthe lens surface is placed within the far field zone of thefeeding source and that the geometrical shape of the lens has acurvature radius larger than the operating wavelength In thisway the travelling electromagneticwave can be approximatedby a set of tubes propagating over a rectilinear path insidethe lens Each tube has a triangular cross section with givensurface area and is defined by three rays departing from eachof the triangle vertices The reflected and transmitted raytubes are determined by using GO principles As a result thepropagation path determines a change of the triangle areathat leads a variation of the electromagnetic field amplitude inaccordance with the power conservation law In Figure 2 thegeometrical change of the tube during the propagation insidethe lens is illustrated In particular the surface area variationfrom 119860119898minus1 to 119860119898 is given by triangle vertices intersectionwith the lens surface in points 119875119860 119875119861 and 119875119862
The surface area of each tube is evaluated by consideringthe overlap between the triangular surface with vertices 119875
119898
119860
119875119898
119861 and 119875
119898
119862 and the elements of the tessellation of the lens
surface The amplitude of the electromagnetic field insidethe tube surface is evaluated by using the Inverse DistanceWeighting (IDW) algorithm In particular considering point119875 inside of the triangle on the lens surface the electric fieldamplitude is given by
E119898119894
(119875) =
E119898119894
(119875119898
119860) 119906119860 + E119898
119894(119875119898
119861) 119906119861 + E119898
119894(119875119898
119862) 119906119862
119906119860 + 119906119861 + 119906119862
(1)
where
E119898119894
(119875119898
119896) = radic
119860119898
119860119898minus1
E119898minus1119894
(119875119898minus1
119896) 119896 = 119860 119861 119862 (2)
119860119898minus1 and 119860119898 denote the area of the triangular sectionof the tube on the lens surface referring to (119898 minus 1)thand 119898th reflection order respectively However in order toaccount for the changes of the tube cross section surfaceand to enforce the energy conservation law the correctioncoefficient radic119860119898119860119898minus1 is introduced In (1) 119875
119898
119860 119875119898
119861 and
119875119898
119862denote the vertices of the surface intersected by the
Mathematical Problems in Engineering 3
tube on the lens surface corresponding to 119898th reflectionorder The coefficients 119906119860 119906119861 and 119906119862 are the inverse of theEuclidean distance between point 119875 and points 119875
119898
119860 119875119898119861 and
119875119898
119862 respectivelyAccording to the GO approach the electric field trans-
mitted outside the lens due to the direct ray and the internalreflections occurring inside the lens is given by the followingequation
E119905 = sum
119898
E119898119905 (3)
where the transmitted field outside the lens due to the 119898thorder internal reflection is
E119898119905
= 119864119898
119905
(n timesk119898
119905) times
k119898
119905
100381710038171003817100381710038171003817(n times
k119898
119905) times
k119898
119905
100381710038171003817100381710038171003817
+ 119864119898
119905perp
n timesk119898
119905
100381710038171003817100381710038171003817n times
k119898
119905
100381710038171003817100381710038171003817
(4)
In (4) n is the unit vector normal to the lens surfacek119898
119905= k119898119905k119898119905 is the transmitted unit wave vector due to
the 119898th internal reflection with k119898119905 = 212058711989901205820 where
1198990 is the refractive index of the external medium and1205820 is the free space wavelength The orthogonal 119864
119898
119905perpand
parallel components 119864119898
119905of the transmitted electric field are
determined bymultiplying the orthogonal119864119898119894perpand parallel119864119898
119894
components of the incident electric field E119898119894in such point by
the proper Fresnel transmission coefficientThe evaluation ofthe incident components of the electric field is carried out bymeans of the following equations
119864119898
119894perp= E119898119894
sdot
n timesk119898
119894
100381710038171003817100381710038171003817n times
k119898
119894
100381710038171003817100381710038171003817
(5)
119864119898
119894= E119898119894
sdot
(n timesk119898
119894) times
k119898
119894
100381710038171003817100381710038171003817(n times
k119898
119894) times
k119898
119894
100381710038171003817100381710038171003817
(6)
where k119898
119894= k119898119894k119898119894 is the incident unit wave vector due
to 119898th internal reflection with k119898119894 = 21205871198991198891205820 where 119899119889 is
the refractive index of the homogeneous dielectric materialforming the lens
In case of direct tube 119898 = 1 the incident field E119898119894(119875119898) at
point 119875119898 is calculated using the far field pattern of the sourceand instead for 119898 gt 1 E119898
119894(119875119898) is derived from (119898 minus 1)th
reflected wave contribution as
E119898119894
(119875119898) = E119898minus1119903
(119875119898minus1) 119890119895k119898119894119889119898
(7)
where 119889119898 is the path length between observation point 119875119898
and point119875119898minus1 where the reflection takes placeThe reflectedfield E119898minus1
119903(119875119898minus1) appearing in (7) is given by the following
equation
E119898minus1119903
(119875119898minus1) = 119864119898minus1
119903
[k119898minus1
119903times (minusn)] times
k119898minus1
119903
100381710038171003817100381710038171003817[k119898minus1
119903times (minusn)] times
k119898minus1
119903
100381710038171003817100381710038171003817
+ 119864119898minus1
119903perp
k119898minus1
119903times (minusn)
100381710038171003817100381710038171003817
k119898minus1
119903times (minusn)
100381710038171003817100381710038171003817
(8)
where parallel 119864119898minus1
119903and orthogonal 119864
119898minus1
119903perpelectric field
components of (119898 minus 1)th reflected wave are obtained bymultiplying the corresponding components with the Fresnelreflection coefficient k
119898
119894= k119898119894k119898119894 term is the reflected unit
vector referring to (119898minus1)th internal reflection at point 119875119898minus1In order to evaluate the transmitted and reflected unit
wave vector Snellrsquos law at the interface between the dielectriclens and the outside medium has to be considered Inparticular the transmission and reflection phenomena aremodeled by the following relations
119899119889k119894 times n = 1198990
k119905 times n
k119894 times n =k119903 times (minusn)
(9)
where the transmitted unit wave vector k119905 the reflected unitwave vector k119903 the tangential unit vector t and the normalunit vector n to the lens surface are given by
k119905 (120579 120601) = cos (120579119905) n + sin (120579119905) t
= [cos (120579119905) 119899119903 + sin (120579119905) 119905119903] r
+ [cos (120579119905) 119899120579 + sin (120579119905) 119905120579]120579
+ [cos (120579119905) 119899120593 + sin (120579119905) 119905120593]
(10)
k119903 times n = (sin 120579119894 minus cos 120579119894) n (11)
t =
minusn times (n timesk119894)
10038171003817100381710038171003817n times (n times
k119894)10038171003817100381710038171003817
(12)
n =
120597r120597120579 times 120597r1205971205931003817100381710038171003817120597r120597120579 times 120597r1205971205931003817
100381710038171003817
(13)
where the normal unit vector n = 119899119903r + 119899120579120579 + 119899120593 and
tangential unit vector t = 119905119903r + 119905120579120579 + 119905120593 to the lens surface
in spherical coordinates are introduced The terms 120579119905 and 120579119894
denote the transmitted and incident angle formed by thewavepropagation vector and the normal unit vector to the surfacein a given point respectively
120579119894 = tanminus1 (k119894 sdot tk119894 sdot n
)
120579119905 = sinminus1 (119899119889 sin 120579119894)
(14)
Once the transmitted and incident angles are computed theFresnel coefficients for the transmitted wave component canbe evaluated as
119905perp =
2119899119889 cos 120579119894119899119889 cos 120579119894 + 1198990 cos 120579119905
119905 =
2119899119889 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119894
(15)
4 Mathematical Problems in Engineering
and the reflected component of the electric field is
119903perp =
119899119889 cos 120579119905 minus 1198990 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119905
119903 =
119899119889 cos 120579119894 minus 1198990 cos 120579119905119899119889 cos 120579119894 + 1198990 cos 120579119905
(16)
Vector r = 119903r appearing in (13) is the radius vector describingthe lens surface defined byGielisrsquo superformula as follows [1718]
r = rradic1199112+ 1199102+ 1199092 (17)
with
119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)
119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)
119911 (120595 120577) = 119877120577 sin (120577)
119877120595 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus11198871
119877120577 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus11198872
(18)
where 119899119901 119886119901 119898119901 119901 = 1 sdot sdot sdot 4 and 119887119902 119902 = 1 2 are real-valuedparameters selected in such a way that the surface of the lensis actually closed and characterized at any point by curvatureradius larger than theworkingwavelength in accordancewiththe GO approximation The parameters 120577 isin [0 1205872] and 120595 isin
[minus120587 120587] denote real-valued parameters appearing in Gielisrsquoformula whereas the spherical angles 120579 and 120593 are obtainedby means of the following relations
120579 = cosminus1 (119911
119903
)
120593 = tanminus1 (119910
119909
)
(19)
The derivatives of the radius vector r appearing in (13) aregiven by
120597r120597120579
=
120597119903
120597120579
r + 119903120579
120597r120597120593
=
120597119903
120597120593
r + 119903 sin (120579)
(20)
where
120597119903
120597120579
=
120597119903
120597120577
120597120577
120597120579
(21)
120597119903
120597120593
=
120597119903
120597120595
minus tan 120574
120597119903
120597120579
(22)
120574 = tanminus1 ( 120597120579
120597120595
) (23)
According to Gielisrsquo description of the lens surface thederivatives of r with respect to 120577 appearing in (21) can beevaluated as follows
120597119903
120597120577
=
1
119903
(119909
120597119909
120597120577
+ 119910
120597119910
120597120577
+ 119911
120597119911
120597120577
)
120597120577
120597120579
=[[
[
119903 (119889119911119889120577) minus 119911 (119889119903119889120577)
1199032radic1 minus (119911119903)
2
]]
]
minus1 (24)
with
119889119911
119889120577
=
119889119877120577
119889120577
sin 120577 + 119877120577 cos 120577
119889119910
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 sin120595 minus 119877120595119877120577 sin 120577 sin120595
119889119909
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 cos120595 minus 119877120595119877120577 sin 120577 cos120595
120597119877120577
120597120577
= minus
1
1198872
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus(1+1198872)1198872
sdot [1198993
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
+ 1198994
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
= minus
1198983
41198863
sin(
1198983120577
4
) [2119867 (1198863) minus 1]
sdot [2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
=
1198984
41198864
cos(1198984120577
4
) [2119867 (1198864) minus 1] [1
minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198984120577
4
100381610038161003816100381610038161003816100381610038162120587
minus 120587)]
(25)
where 119867(sdot) denotes the Heaviside functionThe derivative of rwith respect to120595 appearing in (22) can
be evaluated as follows
120597119903
120597120595
=
1
119903
(119909
120597119909
120597120595
+ 119910
120597119910
120597120595
) (26)
Mathematical Problems in Engineering 5
005
0
minus005
y (m)
x (m)minus002
0
002
0y (m)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0
005
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0 002minus002
x (m)
minus005
0
005
y(m
)
(b)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0 minus005005
y (m)
0
002
004
z(m
)
(c)
minus001minus002 001 0020
x (m)
0
001
002
003
004
005
z(m
)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
(d)
Figure 3 Densities current distribution on the lens surface considering 119898 = 2 internal reflections (a) prospective view (b) top view and (cd) lateral views
where
119889119910
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595
119889119909
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 cos120595 minus 119877120595119877120577 cos 120577 sin120595
120597119877120595
120597120595
= minus
1
1198871
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus(1+1198871)1198871
sdot [1198991
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
+ 1198992
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
= minus
1198981
41198861
sin(
1198981120595
4
) [2119867 (1198861) minus 1]
sdot [2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
=
1198982
41198862
cos(1198982120595
4
) [2119867 (1198862) minus 1] [1
minus 2119867(
1003816100381610038161003816100381610038161003816
1198982120595
4
10038161003816100381610038161003816100381610038162120587
minus 120587)]
120597120579
120597120595
=
119911
1199032[radic1 minus (
119911
119903
)]
minus12
120597119903
120597120595
(27)
Once the contribution of all internal reflections has beenevaluated by the GO method the equivalent electric J119904 andmagnetic M119904 current densities excited along the surface ofthe lens can be easily determined According to the PO
6 Mathematical Problems in Engineering
20
2000
minus20minus20
y
x
z
minus10
0
10
20
30
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
(a)
01020
30
210
60
240
90
270
120
300
150
330
180 0minus10
100∘
120579
120601 = 0∘
120601 = 90∘
(b)
Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90
∘ dashed line of the considered lens antenna
x (m)y (m)
005
0
minus005minus004
minus0020
002 02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0
001
002
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
minus002 0 002minus004
x (m)
minus005
0
005
y(m
)
(b)
0
001
002
z(m
)
0 005minus005
y (m)
020406081
Nor
mal
ized
curr
ent
dens
ities
(c)
0 002minus002minus004
x (m)
0
001
002
z(m
)
020406081
Nor
mal
ized
curr
ent
dens
ities
(d)
Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views
approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression
EFF (119875FF)
= j 119890minusj1198960rFF
21205820rFFint
119878
[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0
119889119878
(28)
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Complex AnalysisJournal of
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
P
PFF
rd
x
rFF
r0
z
ki
120579i
120579t
y
kt
n
Figure 1 Geometrical structure of the electromagnetic systembased on dielectric lens antenna
In this way an improvement of the numerical accuracy canbe obtained Moreover the evaluation of the electromagneticfield radiated from the lens has been carried out applying thephysical optics (PO) approximation In particular accordingto the equivalence principle the radiation outgoing the lensis evaluated by the radiation of the equivalent electric andmagnetic currents on the lens surface The current distribu-tion is calculated by making use of the Fresnel transmissioncoefficient on the lens surface in accordance with the GOprocedure
Thanks to Gielisrsquo formulation it is possible to generate awide range of 3D shapes in a simple and analytical way bychanging a reduced number of parameters The use of thisformulation allows also an easy description of the geometri-cal characteristic of the antenna in terms of lens volume areacurvature radius and aspect ratio Moreover the analyticalrepresentation of the lens shape provides a number of benefitspertaining to the evaluation of the main physical quantitiesinvolved in the electromagnetic propagation equations Inthis way a reduced computation effort is required and moreaccurate results can be obtained
2 Mathematical Model
The geometrical structure considered in the proposed elec-tromagneticmodel consists of a dielectric lens antenna placedon a perfectly electric conductor (PEC) circular plate havingradius 119903119889 (see Figure 1)
The metal plate acts as a ground plane and mechan-ical support of the radiating structure while reducing thebackscattered radiation level The lens is illuminated by theelectromagnetic field emitted by a primary radio source suchas open ended waveguide patch antenna horn antennaor a coaxial probe integrated in the metal plate The tubetracing approach based on the GO approximation is used tomodel the electromagnetic propagation inside the homoge-neous dielectric lensThis approximation allows a significantsimplification of the mathematical model since it makes itpossible to reduce the solution of Maxwellrsquos equations toa geometrical problem As a result the adoption of suchmethod allows the simulation of electrically large structureswith a lower computational effort in comparison to full-wavenumerical methods such as Finite-Difference-Time-Domain(FDTD) and Finite-Element-Method (FEM)
z
y
x
Pmminus1A
Pmminus1B
Pmminus1CAmminus1
PmA
PmB
Am PmC
Figure 2 Area changes from119860119898minus1 to119860119898 of the tube surface duringthe propagation inside the lens
The applicability condition of such method requires thatthe lens surface is placed within the far field zone of thefeeding source and that the geometrical shape of the lens has acurvature radius larger than the operating wavelength In thisway the travelling electromagneticwave can be approximatedby a set of tubes propagating over a rectilinear path insidethe lens Each tube has a triangular cross section with givensurface area and is defined by three rays departing from eachof the triangle vertices The reflected and transmitted raytubes are determined by using GO principles As a result thepropagation path determines a change of the triangle areathat leads a variation of the electromagnetic field amplitude inaccordance with the power conservation law In Figure 2 thegeometrical change of the tube during the propagation insidethe lens is illustrated In particular the surface area variationfrom 119860119898minus1 to 119860119898 is given by triangle vertices intersectionwith the lens surface in points 119875119860 119875119861 and 119875119862
The surface area of each tube is evaluated by consideringthe overlap between the triangular surface with vertices 119875
119898
119860
119875119898
119861 and 119875
119898
119862 and the elements of the tessellation of the lens
surface The amplitude of the electromagnetic field insidethe tube surface is evaluated by using the Inverse DistanceWeighting (IDW) algorithm In particular considering point119875 inside of the triangle on the lens surface the electric fieldamplitude is given by
E119898119894
(119875) =
E119898119894
(119875119898
119860) 119906119860 + E119898
119894(119875119898
119861) 119906119861 + E119898
119894(119875119898
119862) 119906119862
119906119860 + 119906119861 + 119906119862
(1)
where
E119898119894
(119875119898
119896) = radic
119860119898
119860119898minus1
E119898minus1119894
(119875119898minus1
119896) 119896 = 119860 119861 119862 (2)
119860119898minus1 and 119860119898 denote the area of the triangular sectionof the tube on the lens surface referring to (119898 minus 1)thand 119898th reflection order respectively However in order toaccount for the changes of the tube cross section surfaceand to enforce the energy conservation law the correctioncoefficient radic119860119898119860119898minus1 is introduced In (1) 119875
119898
119860 119875119898
119861 and
119875119898
119862denote the vertices of the surface intersected by the
Mathematical Problems in Engineering 3
tube on the lens surface corresponding to 119898th reflectionorder The coefficients 119906119860 119906119861 and 119906119862 are the inverse of theEuclidean distance between point 119875 and points 119875
119898
119860 119875119898119861 and
119875119898
119862 respectivelyAccording to the GO approach the electric field trans-
mitted outside the lens due to the direct ray and the internalreflections occurring inside the lens is given by the followingequation
E119905 = sum
119898
E119898119905 (3)
where the transmitted field outside the lens due to the 119898thorder internal reflection is
E119898119905
= 119864119898
119905
(n timesk119898
119905) times
k119898
119905
100381710038171003817100381710038171003817(n times
k119898
119905) times
k119898
119905
100381710038171003817100381710038171003817
+ 119864119898
119905perp
n timesk119898
119905
100381710038171003817100381710038171003817n times
k119898
119905
100381710038171003817100381710038171003817
(4)
In (4) n is the unit vector normal to the lens surfacek119898
119905= k119898119905k119898119905 is the transmitted unit wave vector due to
the 119898th internal reflection with k119898119905 = 212058711989901205820 where
1198990 is the refractive index of the external medium and1205820 is the free space wavelength The orthogonal 119864
119898
119905perpand
parallel components 119864119898
119905of the transmitted electric field are
determined bymultiplying the orthogonal119864119898119894perpand parallel119864119898
119894
components of the incident electric field E119898119894in such point by
the proper Fresnel transmission coefficientThe evaluation ofthe incident components of the electric field is carried out bymeans of the following equations
119864119898
119894perp= E119898119894
sdot
n timesk119898
119894
100381710038171003817100381710038171003817n times
k119898
119894
100381710038171003817100381710038171003817
(5)
119864119898
119894= E119898119894
sdot
(n timesk119898
119894) times
k119898
119894
100381710038171003817100381710038171003817(n times
k119898
119894) times
k119898
119894
100381710038171003817100381710038171003817
(6)
where k119898
119894= k119898119894k119898119894 is the incident unit wave vector due
to 119898th internal reflection with k119898119894 = 21205871198991198891205820 where 119899119889 is
the refractive index of the homogeneous dielectric materialforming the lens
In case of direct tube 119898 = 1 the incident field E119898119894(119875119898) at
point 119875119898 is calculated using the far field pattern of the sourceand instead for 119898 gt 1 E119898
119894(119875119898) is derived from (119898 minus 1)th
reflected wave contribution as
E119898119894
(119875119898) = E119898minus1119903
(119875119898minus1) 119890119895k119898119894119889119898
(7)
where 119889119898 is the path length between observation point 119875119898
and point119875119898minus1 where the reflection takes placeThe reflectedfield E119898minus1
119903(119875119898minus1) appearing in (7) is given by the following
equation
E119898minus1119903
(119875119898minus1) = 119864119898minus1
119903
[k119898minus1
119903times (minusn)] times
k119898minus1
119903
100381710038171003817100381710038171003817[k119898minus1
119903times (minusn)] times
k119898minus1
119903
100381710038171003817100381710038171003817
+ 119864119898minus1
119903perp
k119898minus1
119903times (minusn)
100381710038171003817100381710038171003817
k119898minus1
119903times (minusn)
100381710038171003817100381710038171003817
(8)
where parallel 119864119898minus1
119903and orthogonal 119864
119898minus1
119903perpelectric field
components of (119898 minus 1)th reflected wave are obtained bymultiplying the corresponding components with the Fresnelreflection coefficient k
119898
119894= k119898119894k119898119894 term is the reflected unit
vector referring to (119898minus1)th internal reflection at point 119875119898minus1In order to evaluate the transmitted and reflected unit
wave vector Snellrsquos law at the interface between the dielectriclens and the outside medium has to be considered Inparticular the transmission and reflection phenomena aremodeled by the following relations
119899119889k119894 times n = 1198990
k119905 times n
k119894 times n =k119903 times (minusn)
(9)
where the transmitted unit wave vector k119905 the reflected unitwave vector k119903 the tangential unit vector t and the normalunit vector n to the lens surface are given by
k119905 (120579 120601) = cos (120579119905) n + sin (120579119905) t
= [cos (120579119905) 119899119903 + sin (120579119905) 119905119903] r
+ [cos (120579119905) 119899120579 + sin (120579119905) 119905120579]120579
+ [cos (120579119905) 119899120593 + sin (120579119905) 119905120593]
(10)
k119903 times n = (sin 120579119894 minus cos 120579119894) n (11)
t =
minusn times (n timesk119894)
10038171003817100381710038171003817n times (n times
k119894)10038171003817100381710038171003817
(12)
n =
120597r120597120579 times 120597r1205971205931003817100381710038171003817120597r120597120579 times 120597r1205971205931003817
100381710038171003817
(13)
where the normal unit vector n = 119899119903r + 119899120579120579 + 119899120593 and
tangential unit vector t = 119905119903r + 119905120579120579 + 119905120593 to the lens surface
in spherical coordinates are introduced The terms 120579119905 and 120579119894
denote the transmitted and incident angle formed by thewavepropagation vector and the normal unit vector to the surfacein a given point respectively
120579119894 = tanminus1 (k119894 sdot tk119894 sdot n
)
120579119905 = sinminus1 (119899119889 sin 120579119894)
(14)
Once the transmitted and incident angles are computed theFresnel coefficients for the transmitted wave component canbe evaluated as
119905perp =
2119899119889 cos 120579119894119899119889 cos 120579119894 + 1198990 cos 120579119905
119905 =
2119899119889 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119894
(15)
4 Mathematical Problems in Engineering
and the reflected component of the electric field is
119903perp =
119899119889 cos 120579119905 minus 1198990 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119905
119903 =
119899119889 cos 120579119894 minus 1198990 cos 120579119905119899119889 cos 120579119894 + 1198990 cos 120579119905
(16)
Vector r = 119903r appearing in (13) is the radius vector describingthe lens surface defined byGielisrsquo superformula as follows [1718]
r = rradic1199112+ 1199102+ 1199092 (17)
with
119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)
119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)
119911 (120595 120577) = 119877120577 sin (120577)
119877120595 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus11198871
119877120577 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus11198872
(18)
where 119899119901 119886119901 119898119901 119901 = 1 sdot sdot sdot 4 and 119887119902 119902 = 1 2 are real-valuedparameters selected in such a way that the surface of the lensis actually closed and characterized at any point by curvatureradius larger than theworkingwavelength in accordancewiththe GO approximation The parameters 120577 isin [0 1205872] and 120595 isin
[minus120587 120587] denote real-valued parameters appearing in Gielisrsquoformula whereas the spherical angles 120579 and 120593 are obtainedby means of the following relations
120579 = cosminus1 (119911
119903
)
120593 = tanminus1 (119910
119909
)
(19)
The derivatives of the radius vector r appearing in (13) aregiven by
120597r120597120579
=
120597119903
120597120579
r + 119903120579
120597r120597120593
=
120597119903
120597120593
r + 119903 sin (120579)
(20)
where
120597119903
120597120579
=
120597119903
120597120577
120597120577
120597120579
(21)
120597119903
120597120593
=
120597119903
120597120595
minus tan 120574
120597119903
120597120579
(22)
120574 = tanminus1 ( 120597120579
120597120595
) (23)
According to Gielisrsquo description of the lens surface thederivatives of r with respect to 120577 appearing in (21) can beevaluated as follows
120597119903
120597120577
=
1
119903
(119909
120597119909
120597120577
+ 119910
120597119910
120597120577
+ 119911
120597119911
120597120577
)
120597120577
120597120579
=[[
[
119903 (119889119911119889120577) minus 119911 (119889119903119889120577)
1199032radic1 minus (119911119903)
2
]]
]
minus1 (24)
with
119889119911
119889120577
=
119889119877120577
119889120577
sin 120577 + 119877120577 cos 120577
119889119910
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 sin120595 minus 119877120595119877120577 sin 120577 sin120595
119889119909
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 cos120595 minus 119877120595119877120577 sin 120577 cos120595
120597119877120577
120597120577
= minus
1
1198872
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus(1+1198872)1198872
sdot [1198993
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
+ 1198994
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
= minus
1198983
41198863
sin(
1198983120577
4
) [2119867 (1198863) minus 1]
sdot [2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
=
1198984
41198864
cos(1198984120577
4
) [2119867 (1198864) minus 1] [1
minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198984120577
4
100381610038161003816100381610038161003816100381610038162120587
minus 120587)]
(25)
where 119867(sdot) denotes the Heaviside functionThe derivative of rwith respect to120595 appearing in (22) can
be evaluated as follows
120597119903
120597120595
=
1
119903
(119909
120597119909
120597120595
+ 119910
120597119910
120597120595
) (26)
Mathematical Problems in Engineering 5
005
0
minus005
y (m)
x (m)minus002
0
002
0y (m)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0
005
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0 002minus002
x (m)
minus005
0
005
y(m
)
(b)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0 minus005005
y (m)
0
002
004
z(m
)
(c)
minus001minus002 001 0020
x (m)
0
001
002
003
004
005
z(m
)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
(d)
Figure 3 Densities current distribution on the lens surface considering 119898 = 2 internal reflections (a) prospective view (b) top view and (cd) lateral views
where
119889119910
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595
119889119909
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 cos120595 minus 119877120595119877120577 cos 120577 sin120595
120597119877120595
120597120595
= minus
1
1198871
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus(1+1198871)1198871
sdot [1198991
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
+ 1198992
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
= minus
1198981
41198861
sin(
1198981120595
4
) [2119867 (1198861) minus 1]
sdot [2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
=
1198982
41198862
cos(1198982120595
4
) [2119867 (1198862) minus 1] [1
minus 2119867(
1003816100381610038161003816100381610038161003816
1198982120595
4
10038161003816100381610038161003816100381610038162120587
minus 120587)]
120597120579
120597120595
=
119911
1199032[radic1 minus (
119911
119903
)]
minus12
120597119903
120597120595
(27)
Once the contribution of all internal reflections has beenevaluated by the GO method the equivalent electric J119904 andmagnetic M119904 current densities excited along the surface ofthe lens can be easily determined According to the PO
6 Mathematical Problems in Engineering
20
2000
minus20minus20
y
x
z
minus10
0
10
20
30
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
(a)
01020
30
210
60
240
90
270
120
300
150
330
180 0minus10
100∘
120579
120601 = 0∘
120601 = 90∘
(b)
Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90
∘ dashed line of the considered lens antenna
x (m)y (m)
005
0
minus005minus004
minus0020
002 02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0
001
002
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
minus002 0 002minus004
x (m)
minus005
0
005
y(m
)
(b)
0
001
002
z(m
)
0 005minus005
y (m)
020406081
Nor
mal
ized
curr
ent
dens
ities
(c)
0 002minus002minus004
x (m)
0
001
002
z(m
)
020406081
Nor
mal
ized
curr
ent
dens
ities
(d)
Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views
approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression
EFF (119875FF)
= j 119890minusj1198960rFF
21205820rFFint
119878
[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0
119889119878
(28)
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
tube on the lens surface corresponding to 119898th reflectionorder The coefficients 119906119860 119906119861 and 119906119862 are the inverse of theEuclidean distance between point 119875 and points 119875
119898
119860 119875119898119861 and
119875119898
119862 respectivelyAccording to the GO approach the electric field trans-
mitted outside the lens due to the direct ray and the internalreflections occurring inside the lens is given by the followingequation
E119905 = sum
119898
E119898119905 (3)
where the transmitted field outside the lens due to the 119898thorder internal reflection is
E119898119905
= 119864119898
119905
(n timesk119898
119905) times
k119898
119905
100381710038171003817100381710038171003817(n times
k119898
119905) times
k119898
119905
100381710038171003817100381710038171003817
+ 119864119898
119905perp
n timesk119898
119905
100381710038171003817100381710038171003817n times
k119898
119905
100381710038171003817100381710038171003817
(4)
In (4) n is the unit vector normal to the lens surfacek119898
119905= k119898119905k119898119905 is the transmitted unit wave vector due to
the 119898th internal reflection with k119898119905 = 212058711989901205820 where
1198990 is the refractive index of the external medium and1205820 is the free space wavelength The orthogonal 119864
119898
119905perpand
parallel components 119864119898
119905of the transmitted electric field are
determined bymultiplying the orthogonal119864119898119894perpand parallel119864119898
119894
components of the incident electric field E119898119894in such point by
the proper Fresnel transmission coefficientThe evaluation ofthe incident components of the electric field is carried out bymeans of the following equations
119864119898
119894perp= E119898119894
sdot
n timesk119898
119894
100381710038171003817100381710038171003817n times
k119898
119894
100381710038171003817100381710038171003817
(5)
119864119898
119894= E119898119894
sdot
(n timesk119898
119894) times
k119898
119894
100381710038171003817100381710038171003817(n times
k119898
119894) times
k119898
119894
100381710038171003817100381710038171003817
(6)
where k119898
119894= k119898119894k119898119894 is the incident unit wave vector due
to 119898th internal reflection with k119898119894 = 21205871198991198891205820 where 119899119889 is
the refractive index of the homogeneous dielectric materialforming the lens
In case of direct tube 119898 = 1 the incident field E119898119894(119875119898) at
point 119875119898 is calculated using the far field pattern of the sourceand instead for 119898 gt 1 E119898
119894(119875119898) is derived from (119898 minus 1)th
reflected wave contribution as
E119898119894
(119875119898) = E119898minus1119903
(119875119898minus1) 119890119895k119898119894119889119898
(7)
where 119889119898 is the path length between observation point 119875119898
and point119875119898minus1 where the reflection takes placeThe reflectedfield E119898minus1
119903(119875119898minus1) appearing in (7) is given by the following
equation
E119898minus1119903
(119875119898minus1) = 119864119898minus1
119903
[k119898minus1
119903times (minusn)] times
k119898minus1
119903
100381710038171003817100381710038171003817[k119898minus1
119903times (minusn)] times
k119898minus1
119903
100381710038171003817100381710038171003817
+ 119864119898minus1
119903perp
k119898minus1
119903times (minusn)
100381710038171003817100381710038171003817
k119898minus1
119903times (minusn)
100381710038171003817100381710038171003817
(8)
where parallel 119864119898minus1
119903and orthogonal 119864
119898minus1
119903perpelectric field
components of (119898 minus 1)th reflected wave are obtained bymultiplying the corresponding components with the Fresnelreflection coefficient k
119898
119894= k119898119894k119898119894 term is the reflected unit
vector referring to (119898minus1)th internal reflection at point 119875119898minus1In order to evaluate the transmitted and reflected unit
wave vector Snellrsquos law at the interface between the dielectriclens and the outside medium has to be considered Inparticular the transmission and reflection phenomena aremodeled by the following relations
119899119889k119894 times n = 1198990
k119905 times n
k119894 times n =k119903 times (minusn)
(9)
where the transmitted unit wave vector k119905 the reflected unitwave vector k119903 the tangential unit vector t and the normalunit vector n to the lens surface are given by
k119905 (120579 120601) = cos (120579119905) n + sin (120579119905) t
= [cos (120579119905) 119899119903 + sin (120579119905) 119905119903] r
+ [cos (120579119905) 119899120579 + sin (120579119905) 119905120579]120579
+ [cos (120579119905) 119899120593 + sin (120579119905) 119905120593]
(10)
k119903 times n = (sin 120579119894 minus cos 120579119894) n (11)
t =
minusn times (n timesk119894)
10038171003817100381710038171003817n times (n times
k119894)10038171003817100381710038171003817
(12)
n =
120597r120597120579 times 120597r1205971205931003817100381710038171003817120597r120597120579 times 120597r1205971205931003817
100381710038171003817
(13)
where the normal unit vector n = 119899119903r + 119899120579120579 + 119899120593 and
tangential unit vector t = 119905119903r + 119905120579120579 + 119905120593 to the lens surface
in spherical coordinates are introduced The terms 120579119905 and 120579119894
denote the transmitted and incident angle formed by thewavepropagation vector and the normal unit vector to the surfacein a given point respectively
120579119894 = tanminus1 (k119894 sdot tk119894 sdot n
)
120579119905 = sinminus1 (119899119889 sin 120579119894)
(14)
Once the transmitted and incident angles are computed theFresnel coefficients for the transmitted wave component canbe evaluated as
119905perp =
2119899119889 cos 120579119894119899119889 cos 120579119894 + 1198990 cos 120579119905
119905 =
2119899119889 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119894
(15)
4 Mathematical Problems in Engineering
and the reflected component of the electric field is
119903perp =
119899119889 cos 120579119905 minus 1198990 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119905
119903 =
119899119889 cos 120579119894 minus 1198990 cos 120579119905119899119889 cos 120579119894 + 1198990 cos 120579119905
(16)
Vector r = 119903r appearing in (13) is the radius vector describingthe lens surface defined byGielisrsquo superformula as follows [1718]
r = rradic1199112+ 1199102+ 1199092 (17)
with
119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)
119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)
119911 (120595 120577) = 119877120577 sin (120577)
119877120595 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus11198871
119877120577 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus11198872
(18)
where 119899119901 119886119901 119898119901 119901 = 1 sdot sdot sdot 4 and 119887119902 119902 = 1 2 are real-valuedparameters selected in such a way that the surface of the lensis actually closed and characterized at any point by curvatureradius larger than theworkingwavelength in accordancewiththe GO approximation The parameters 120577 isin [0 1205872] and 120595 isin
[minus120587 120587] denote real-valued parameters appearing in Gielisrsquoformula whereas the spherical angles 120579 and 120593 are obtainedby means of the following relations
120579 = cosminus1 (119911
119903
)
120593 = tanminus1 (119910
119909
)
(19)
The derivatives of the radius vector r appearing in (13) aregiven by
120597r120597120579
=
120597119903
120597120579
r + 119903120579
120597r120597120593
=
120597119903
120597120593
r + 119903 sin (120579)
(20)
where
120597119903
120597120579
=
120597119903
120597120577
120597120577
120597120579
(21)
120597119903
120597120593
=
120597119903
120597120595
minus tan 120574
120597119903
120597120579
(22)
120574 = tanminus1 ( 120597120579
120597120595
) (23)
According to Gielisrsquo description of the lens surface thederivatives of r with respect to 120577 appearing in (21) can beevaluated as follows
120597119903
120597120577
=
1
119903
(119909
120597119909
120597120577
+ 119910
120597119910
120597120577
+ 119911
120597119911
120597120577
)
120597120577
120597120579
=[[
[
119903 (119889119911119889120577) minus 119911 (119889119903119889120577)
1199032radic1 minus (119911119903)
2
]]
]
minus1 (24)
with
119889119911
119889120577
=
119889119877120577
119889120577
sin 120577 + 119877120577 cos 120577
119889119910
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 sin120595 minus 119877120595119877120577 sin 120577 sin120595
119889119909
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 cos120595 minus 119877120595119877120577 sin 120577 cos120595
120597119877120577
120597120577
= minus
1
1198872
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus(1+1198872)1198872
sdot [1198993
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
+ 1198994
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
= minus
1198983
41198863
sin(
1198983120577
4
) [2119867 (1198863) minus 1]
sdot [2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
=
1198984
41198864
cos(1198984120577
4
) [2119867 (1198864) minus 1] [1
minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198984120577
4
100381610038161003816100381610038161003816100381610038162120587
minus 120587)]
(25)
where 119867(sdot) denotes the Heaviside functionThe derivative of rwith respect to120595 appearing in (22) can
be evaluated as follows
120597119903
120597120595
=
1
119903
(119909
120597119909
120597120595
+ 119910
120597119910
120597120595
) (26)
Mathematical Problems in Engineering 5
005
0
minus005
y (m)
x (m)minus002
0
002
0y (m)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0
005
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0 002minus002
x (m)
minus005
0
005
y(m
)
(b)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0 minus005005
y (m)
0
002
004
z(m
)
(c)
minus001minus002 001 0020
x (m)
0
001
002
003
004
005
z(m
)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
(d)
Figure 3 Densities current distribution on the lens surface considering 119898 = 2 internal reflections (a) prospective view (b) top view and (cd) lateral views
where
119889119910
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595
119889119909
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 cos120595 minus 119877120595119877120577 cos 120577 sin120595
120597119877120595
120597120595
= minus
1
1198871
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus(1+1198871)1198871
sdot [1198991
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
+ 1198992
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
= minus
1198981
41198861
sin(
1198981120595
4
) [2119867 (1198861) minus 1]
sdot [2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
=
1198982
41198862
cos(1198982120595
4
) [2119867 (1198862) minus 1] [1
minus 2119867(
1003816100381610038161003816100381610038161003816
1198982120595
4
10038161003816100381610038161003816100381610038162120587
minus 120587)]
120597120579
120597120595
=
119911
1199032[radic1 minus (
119911
119903
)]
minus12
120597119903
120597120595
(27)
Once the contribution of all internal reflections has beenevaluated by the GO method the equivalent electric J119904 andmagnetic M119904 current densities excited along the surface ofthe lens can be easily determined According to the PO
6 Mathematical Problems in Engineering
20
2000
minus20minus20
y
x
z
minus10
0
10
20
30
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
(a)
01020
30
210
60
240
90
270
120
300
150
330
180 0minus10
100∘
120579
120601 = 0∘
120601 = 90∘
(b)
Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90
∘ dashed line of the considered lens antenna
x (m)y (m)
005
0
minus005minus004
minus0020
002 02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0
001
002
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
minus002 0 002minus004
x (m)
minus005
0
005
y(m
)
(b)
0
001
002
z(m
)
0 005minus005
y (m)
020406081
Nor
mal
ized
curr
ent
dens
ities
(c)
0 002minus002minus004
x (m)
0
001
002
z(m
)
020406081
Nor
mal
ized
curr
ent
dens
ities
(d)
Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views
approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression
EFF (119875FF)
= j 119890minusj1198960rFF
21205820rFFint
119878
[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0
119889119878
(28)
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
and the reflected component of the electric field is
119903perp =
119899119889 cos 120579119905 minus 1198990 cos 120579119894119899119889 cos 120579119905 + 1198990 cos 120579119905
119903 =
119899119889 cos 120579119894 minus 1198990 cos 120579119905119899119889 cos 120579119894 + 1198990 cos 120579119905
(16)
Vector r = 119903r appearing in (13) is the radius vector describingthe lens surface defined byGielisrsquo superformula as follows [1718]
r = rradic1199112+ 1199102+ 1199092 (17)
with
119909 (120595 120577) = 119877120595 cos (120595) 119877120577 cos (120577)
119910 (120595 120577) = 119877120595 sin (120595) 119877120577 cos (120577)
119911 (120595 120577) = 119877120577 sin (120577)
119877120595 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus11198871
119877120577 = [
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus11198872
(18)
where 119899119901 119886119901 119898119901 119901 = 1 sdot sdot sdot 4 and 119887119902 119902 = 1 2 are real-valuedparameters selected in such a way that the surface of the lensis actually closed and characterized at any point by curvatureradius larger than theworkingwavelength in accordancewiththe GO approximation The parameters 120577 isin [0 1205872] and 120595 isin
[minus120587 120587] denote real-valued parameters appearing in Gielisrsquoformula whereas the spherical angles 120579 and 120593 are obtainedby means of the following relations
120579 = cosminus1 (119911
119903
)
120593 = tanminus1 (119910
119909
)
(19)
The derivatives of the radius vector r appearing in (13) aregiven by
120597r120597120579
=
120597119903
120597120579
r + 119903120579
120597r120597120593
=
120597119903
120597120593
r + 119903 sin (120579)
(20)
where
120597119903
120597120579
=
120597119903
120597120577
120597120577
120597120579
(21)
120597119903
120597120593
=
120597119903
120597120595
minus tan 120574
120597119903
120597120579
(22)
120574 = tanminus1 ( 120597120579
120597120595
) (23)
According to Gielisrsquo description of the lens surface thederivatives of r with respect to 120577 appearing in (21) can beevaluated as follows
120597119903
120597120577
=
1
119903
(119909
120597119909
120597120577
+ 119910
120597119910
120597120577
+ 119911
120597119911
120597120577
)
120597120577
120597120579
=[[
[
119903 (119889119911119889120577) minus 119911 (119889119903119889120577)
1199032radic1 minus (119911119903)
2
]]
]
minus1 (24)
with
119889119911
119889120577
=
119889119877120577
119889120577
sin 120577 + 119877120577 cos 120577
119889119910
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 sin120595 minus 119877120595119877120577 sin 120577 sin120595
119889119909
119889120577
= 119877120595
119889119877120577
119889120577
cos 120577 cos120595 minus 119877120595119877120577 sin 120577 cos120595
120597119877120577
120597120577
= minus
1
1198872
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994
]
minus(1+1198872)1198872
sdot [1198993
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
1198993minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
+ 1198994
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
1198994minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989831205774)1198863
100381610038161003816100381610038161003816100381610038161003816
= minus
1198983
41198863
sin(
1198983120577
4
) [2119867 (1198863) minus 1]
sdot [2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198983120577
4
100381610038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989841205774)
1198864
100381610038161003816100381610038161003816100381610038161003816
=
1198984
41198864
cos(1198984120577
4
) [2119867 (1198864) minus 1] [1
minus 2119867(
10038161003816100381610038161003816100381610038161003816
1198984120577
4
100381610038161003816100381610038161003816100381610038162120587
minus 120587)]
(25)
where 119867(sdot) denotes the Heaviside functionThe derivative of rwith respect to120595 appearing in (22) can
be evaluated as follows
120597119903
120597120595
=
1
119903
(119909
120597119909
120597120595
+ 119910
120597119910
120597120595
) (26)
Mathematical Problems in Engineering 5
005
0
minus005
y (m)
x (m)minus002
0
002
0y (m)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0
005
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0 002minus002
x (m)
minus005
0
005
y(m
)
(b)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0 minus005005
y (m)
0
002
004
z(m
)
(c)
minus001minus002 001 0020
x (m)
0
001
002
003
004
005
z(m
)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
(d)
Figure 3 Densities current distribution on the lens surface considering 119898 = 2 internal reflections (a) prospective view (b) top view and (cd) lateral views
where
119889119910
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595
119889119909
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 cos120595 minus 119877120595119877120577 cos 120577 sin120595
120597119877120595
120597120595
= minus
1
1198871
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus(1+1198871)1198871
sdot [1198991
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
+ 1198992
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
= minus
1198981
41198861
sin(
1198981120595
4
) [2119867 (1198861) minus 1]
sdot [2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
=
1198982
41198862
cos(1198982120595
4
) [2119867 (1198862) minus 1] [1
minus 2119867(
1003816100381610038161003816100381610038161003816
1198982120595
4
10038161003816100381610038161003816100381610038162120587
minus 120587)]
120597120579
120597120595
=
119911
1199032[radic1 minus (
119911
119903
)]
minus12
120597119903
120597120595
(27)
Once the contribution of all internal reflections has beenevaluated by the GO method the equivalent electric J119904 andmagnetic M119904 current densities excited along the surface ofthe lens can be easily determined According to the PO
6 Mathematical Problems in Engineering
20
2000
minus20minus20
y
x
z
minus10
0
10
20
30
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
(a)
01020
30
210
60
240
90
270
120
300
150
330
180 0minus10
100∘
120579
120601 = 0∘
120601 = 90∘
(b)
Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90
∘ dashed line of the considered lens antenna
x (m)y (m)
005
0
minus005minus004
minus0020
002 02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0
001
002
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
minus002 0 002minus004
x (m)
minus005
0
005
y(m
)
(b)
0
001
002
z(m
)
0 005minus005
y (m)
020406081
Nor
mal
ized
curr
ent
dens
ities
(c)
0 002minus002minus004
x (m)
0
001
002
z(m
)
020406081
Nor
mal
ized
curr
ent
dens
ities
(d)
Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views
approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression
EFF (119875FF)
= j 119890minusj1198960rFF
21205820rFFint
119878
[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0
119889119878
(28)
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
005
0
minus005
y (m)
x (m)minus002
0
002
0y (m)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0
005
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
0 002minus002
x (m)
minus005
0
005
y(m
)
(b)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0 minus005005
y (m)
0
002
004
z(m
)
(c)
minus001minus002 001 0020
x (m)
0
001
002
003
004
005
z(m
)
02
04
06
08
1
Nor
mal
ized
curr
ent d
ensit
ies
(d)
Figure 3 Densities current distribution on the lens surface considering 119898 = 2 internal reflections (a) prospective view (b) top view and (cd) lateral views
where
119889119910
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 sin120595 + 119877120595119877120577 cos 120577 cos120595
119889119909
119889120595
= 119877120577
119889119877120595
119889120595
cos 120577 cos120595 minus 119877120595119877120577 cos 120577 sin120595
120597119877120595
120597120595
= minus
1
1198871
[
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991
+
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992
]
minus(1+1198871)1198871
sdot [1198991
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
1198991minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
+ 1198992
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
1198992minus1
120597
120597120577
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
cos (11989811205954)
1198861
100381610038161003816100381610038161003816100381610038161003816
= minus
1198981
41198861
sin(
1198981120595
4
) [2119867 (1198861) minus 1]
sdot [2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
3
4
120587) minus 2119867(
1003816100381610038161003816100381610038161003816
1198981120595
4
10038161003816100381610038161003816100381610038162120587
minus
120587
2
)
+ 1]
120597
120597120595
100381610038161003816100381610038161003816100381610038161003816
sin (11989821205954)
1198862
100381610038161003816100381610038161003816100381610038161003816
=
1198982
41198862
cos(1198982120595
4
) [2119867 (1198862) minus 1] [1
minus 2119867(
1003816100381610038161003816100381610038161003816
1198982120595
4
10038161003816100381610038161003816100381610038162120587
minus 120587)]
120597120579
120597120595
=
119911
1199032[radic1 minus (
119911
119903
)]
minus12
120597119903
120597120595
(27)
Once the contribution of all internal reflections has beenevaluated by the GO method the equivalent electric J119904 andmagnetic M119904 current densities excited along the surface ofthe lens can be easily determined According to the PO
6 Mathematical Problems in Engineering
20
2000
minus20minus20
y
x
z
minus10
0
10
20
30
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
(a)
01020
30
210
60
240
90
270
120
300
150
330
180 0minus10
100∘
120579
120601 = 0∘
120601 = 90∘
(b)
Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90
∘ dashed line of the considered lens antenna
x (m)y (m)
005
0
minus005minus004
minus0020
002 02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0
001
002
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
minus002 0 002minus004
x (m)
minus005
0
005
y(m
)
(b)
0
001
002
z(m
)
0 005minus005
y (m)
020406081
Nor
mal
ized
curr
ent
dens
ities
(c)
0 002minus002minus004
x (m)
0
001
002
z(m
)
020406081
Nor
mal
ized
curr
ent
dens
ities
(d)
Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views
approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression
EFF (119875FF)
= j 119890minusj1198960rFF
21205820rFFint
119878
[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0
119889119878
(28)
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
20
2000
minus20minus20
y
x
z
minus10
0
10
20
30
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
(a)
01020
30
210
60
240
90
270
120
300
150
330
180 0minus10
100∘
120579
120601 = 0∘
120601 = 90∘
(b)
Figure 4 Radiation solid (a) and polar sections (b) with 120601 = 0∘ solid line and 120601 = 90
∘ dashed line of the considered lens antenna
x (m)y (m)
005
0
minus005minus004
minus0020
002 02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
0
001
002
z(m
)
(a)
02
04
06
08
1
Nor
mal
ized
curr
ent
dens
ities
minus002 0 002minus004
x (m)
minus005
0
005
y(m
)
(b)
0
001
002
z(m
)
0 005minus005
y (m)
020406081
Nor
mal
ized
curr
ent
dens
ities
(c)
0 002minus002minus004
x (m)
0
001
002
z(m
)
020406081
Nor
mal
ized
curr
ent
dens
ities
(d)
Figure 5 Densities current distribution on the lens surface considering 119898 = 3 internal reflection (a) prospective view (b) top view and (cd) lateral views
approximation the electromagnetic field radiated in thespatial domain outside the lens at the general observationpoint 119875FF = (rFF 120579FF 120593FF) can be computed by the followingintegral expression
EFF (119875FF)
= j 119890minusj1198960rFF
21205820rFFint
119878
[1205780J119878 times u0 minus M119878] times u0119890j1198960r0sdotu0
119889119878
(28)
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
z
2020
0 0
minus20 minus20
yx
minus20
minus15
minus10
minus5
0
5
10
Dire
ctiv
ity (d
Bi)
minus10
0
10
20
30
(a)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0120579
120601 = 0∘
120601 = 90∘
minus10
70∘
(b)
0
10
20
30
210
60
240
90
270
120
300
150
330
180 0
120601 = 0∘
120601 = 60∘
120601 = 120∘
120579
minus10
(c)
Figure 6 (a) Radiation solid (b) polar sections with 120601 = 0∘ (solid line) and 120601 = 90
∘ (dashed line) and (c) polar sections of the three mainlobes in 120601 = 0
∘ (solid line) 120601 = 60∘ (dashed line) and 120601 = 120
∘ (dotted line)
where r0 is the vector pointing from point 119875 on the lenssurface and observation point 119875FF 1205780 is the characteristicimpedance of the vacuum rFF is the distance vector betweenthe origin of the coordinate system and observation point119875FFand u0 is the unit vector corresponding to r In such way thedirectivity of the antenna can be obtained by the followingexpression
ℎ (120579FF 120593FF) =
4120587r2FF1003817100381710038171003817EFF
1003817100381710038171003817
2
1205780119875tot (29)
where 119875tot is the total electromagnetic power radiated by thelens
3 Numerical Results
Gielisrsquo formulation allows the design of the proper lensshape by changing a few equation parameters In this wayit is possible to analytically define the lens geometry usefulfor meeting size constraints due to the packaging round-ness aspect ratio or any sort of shape-related requirement
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(a)
30
210
60
240
90
270
120
300
150
330
180 0
Full waveGOPO approach
0
minus10
minus20
minus30
minus40
(b)
Figure 7 Comparison between the normalized directivity of the LA1 lens antenna polar sections with (a) 120601 = 0∘ and (b) 120601 = 90
∘
Furthermore the electromagnetic radiation properties can betuned by changing the lens parameters in order to synthesizea desired radiation pattern Moreover Gielisrsquo superformulais useful for defining analytically all the physical quantitiesinvolved in the electromagnetic problem This in turn isbeneficial for enhancing the accuracy of the results and at thesame time reducing the computational cost of the analysis incomparison to a brute-force numerical approach
The developedmodeling technique is adopted to design aparticular lens antenna showing a flat-top radiating patternat frequency 119891 = 60GHz This type of antenna couldbe integrated in communication systems covering a widearea The proposed lens (LA1) is made out of a dielectricmaterial with refractive index equal to 119899119889 = 142 havinga minimum radius of 119903min = 25mm and described bythe following parameters of Gielisrsquo superformula 119899119894 = 4119886119894 = 1 119898119894 = 2 119894 = 1 sdot sdot sdot 4 and 1198871 = 1198872 = 1 Theselected Gielisrsquo parameters ensure a curvature radius of thelens larger than the operating wavelength in all the pointsalong the lens surface in accordance with the applicabilityrestrictions of the GO approximation The lens is placed ona metal disk plate with radius 119903119889 = 100mm and fed by anopen ended circular waveguide with diameter 119886119892 = 23mmfilled up by the same dielectric material forming the lens andpositioned at center of such metallic plate The numericalsimulation has been performed considering a lens surfacetessellation composed by 160 times 160 elements and 119898 = 2
internal reflections Figure 3 shows the equivalent electriccurrent densities distribution on the lens surface consideringthe internal reflection contribution
Figure 4 illustrates (a) the radiation solids (far fieldpattern) and (b) the polar sections generated by the densitiescurrent pattern illustrated in Figure 3 It is worth noting thatthe antennamain lobe is characterized by an angular aperture
at minus3 dB of about 100 degrees with a peak directivity value ofabout 13 dBi
Another numerical example relevant to the design of alens antenna characterized by a triple radiation beam patternat frequency 119891 = 60GHz is here reported This type ofantenna could be adopted in communication systems imple-menting a spatial-division multiplexing useful for increasingthe channel capacity where the position of the receiver isknown In this case the proposed lens (LA2) is made out ofa dielectric material with refractive index equal to 119899119889 = 142having a minimum radius of 119903min = 25mm and described bythe following Gielisrsquo parameters 1198991 = 1198992 = 4 1198993 = 3 1198994 = 1119886119894 = 1 and 119898119894 = 3 (119894 = 1 sdot sdot sdot 4) Also in this case the selectedGielisrsquo parameters ensure a minimum curvature radius ofthe lens surface that satisfies the applicability conditions ofthe GO approximation A metal disk plate with radius 119903119889 =
100mm is also considered and the feeding source consistsof an open ended circular waveguide having diameter 119886119892 =
27mm and filled up by the same dielectric material formingthe lens Also in this case the lens surface tessellation iscomposed by 160 times 160 elements and 119898 = 3 internalreflections have been taken into account In Figure 5 thedistribution of the equivalent electric current densities on thelens surface is shown The internal reflections are properlytaken into account
Figure 6 shows (a) the radiation solids and ((b) (c))polar sections generated by the current density distributionillustrated in Figure 5 It is possible to observe three mainlobes having an angular aperture at minus3 dB of about 70 degreespointing in three different directions 120601 = 0
∘ 120601 = 120∘ and
120601 = 240∘ with a directivity of about 11 dBi and an elevation
angle of 120579 = 45∘
Both the illustrated lens antenna structures feature spe-cific properties in terms of radiation pattern that can be
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(a)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(b)
30
210
60
240
90
270
120
300
150
330
180 0
0
minus10
minus20
minus30
minus40
Full waveGOPO approach
(c)
Figure 8 Comparison between the normalized directivity of the LA2 lens antenna polar sections with (a) 120601 = 0∘ (b) 120601 = 60
∘ and (c)120601 = 120
beneficial for the newly introducedWi-Fi 80211ad communi-cation protocol working at frequency119891 = 60GHzMoreoverthe obtained numerical results confirm the feasibility of theproposed mathematical model to study the special class oflens antennas defined by Gielisrsquo superformula
4 Model Validation
In order to validate the developed modeling approach acomparison with a commercially available full-wave electro-magnetic solver CSTMicrowave Studio has been carried out
Figures 7 and 8 show the normalized directivity of the lensantennas LA1 and LA2 respectively as computed by thedeveloped GOPO asymptotic approach and the commercialsoftware It is worthwhile to note that within the angularwidth of the main lobe (antenna beam width or half-powerbeam width more important for antenna design) a closeragreement between the full-wave and GO-PO numericalresults has been obtained Outside such angular region (lessimportant for antenna design) an accuracy decrease of theGO-PO results can be observed This occurrence can beexplained taking into account that the source field used in
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
GO-PO approach is slightly different from that resultingfrom the full-wave method In fact in GO-PO approachthe incident electromagnetic field on the lens surface corre-sponds to the far field pattern from a circular aperture onan infinite metallic plate Instead in the full-wave methodthe calculation of the incident electromagnetic field takesinto account the finite size of the metallic plate Howeverthe full-wave analysis requires more computational resourcesin terms of memory occupation and simulation time As amatter of fact using aworkstationwith dual Intel Xeon E5645processor frequency of 24GHz the computational time andmemory allocation required by the full-wave solver are about15 hr and 22GBytes respectively for lens LA1 and about10 hr and 15GBytes respectively for lens LA2 On the otherhand the developed GO-PO procedure is characterized by acomputational time andmemory allocation of about 3 hr and25GBytes on the same workstation
5 Conclusion
A novel mathematical procedure based on GOPO tube trac-ing approach for modeling the electromagnetic propagationinside the lens antenna has been illustrated The proposedmodel is applied to a special class of lens antennas havingthe three-dimensional shape defined by Gielisrsquo superformulaThe analytical expression of the lens surface reduces thedrawbacks due to various numerical approximations whileensuring more accurate results The applicability of theproposedmodel could be extended to design complex shapedlens antennas for a wide range of applications covering boththe microwave and optical frequency band A proper selec-tion of Gielisrsquo shape allows the performance improvementof the antenna in terms of directivity multibeam radiatingshape beam-steering angle beam stability wide beam angleand radiating efficiency
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Raman N S Barker andGM Rebeiz ldquoAW-band dielectric-lens-based integratedmonopulse radar receiverrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 46 no 12 pp2308ndash2316 1998
[2] A Bisognin D Titz F Ferrero et al ldquo3D printed plastic 60GHzlens enabling innovative millimeter wave antenna solutionand systemrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium (IMS rsquo14) pp 1ndash4 Tampa Fla USA June2014
[3] K Uehara K Miyashita K-I Natsume K Hatakeyama and KMizuno ldquoLens-coupled imaging arrays for the millimeter- andsubmillimeter-wave regionsrdquo IEEE Transactions on MicrowaveTheory and Techniques vol 40 no 5 pp 806ndash811 1992
[4] O Yurduseven N Llombart A Neto and J Baselmans ldquoA dualpolarized antenna for THz space applications antenna designand lens optimizationrdquo in Proceedings of the IEEE Antennas and
Propagation Society International Symposium (APSURSI rsquo14) pp191ndash192 IEEE Memphis Tenn USA July 2014
[5] S Ravishankar and B Dharshak ldquoA rapid direction of arrivalestimation procedure for adaptive array antennas covered by ashaped dielectric lensrdquo in Proceedings of the IEEE Radio andWireless Symposium (RWS rsquo14) pp 124ndash126 Newport BeachCalif USA January 2014
[6] T H Buttgenbach ldquoAn improved solution for integrated arrayoptics in quasi-optical mm and submm receivers the hybridantennardquo IEEE Transactions on Microwave Theory and Tech-niques vol 41 no 10 pp 1750ndash1761 1993
[7] D F Filipovic and G M Rebeiz ldquoDouble-slot antennas onextended hemispherical and elliptical quartz dielectric lensesrdquoInternational Journal of Infrared and Millimeter Waves vol 14no 10 pp 1905ndash1924 1993
[8] J J LeeHandbook of Microwave and Optical Components JohnWiley amp Sons New York NY USA 1989
[9] B Chantraine-Bares R Sauleau L Le Coq and K MahdjoubildquoA new accurate design method for millimeter-wave homo-geneous dielectric substrate lens antennas of arbitrary shaperdquoIEEE Transactions on Antennas and Propagation vol 53 no 3pp 1069ndash1082 2005
[10] T Dang J Yang and H-X Zheng ldquoAn integrated lens antennadesign with irregular lens profilerdquo in Proceedings of the 5thGlobal Symposium on Millimeter-Waves (GSMM rsquo12) pp 212ndash215 Harbin China May 2012
[11] T Jaschke B Rohrdantz and A F Jacob ldquoA flexible surfacedescription for arbitrarily shaped dielectric lens antennasrdquo inProceedings of the German Microwave Conference (GeMIC rsquo14)pp 1ndash4 Aachen Germany March 2014
[12] A P Pavacic D L del Rıo J R Mosig and G V EleftheriadesldquoThree-dimensional ray-tracing to model internal reflectionsin off-axis lens antennasrdquo IEEE Transactions on Antennas andPropagation vol 54 no 2 pp 604ndash612 2006
[13] A Neto S Maci and P J I de Maagt ldquoReflections inside anelliptical dielectric lens antennardquo IEE ProceedingsmdashMicrowavesAntennas and Propagation vol 145 no 3 pp 243ndash247 1998
[14] J Gielis ldquoA generic geometric transformation that unifies awide range of natural and abstract shapesrdquo American Journalof Botany vol 90 no 3 pp 333ndash338 2003
[15] M Simeoni R Cicchetti A Yarovoy and D Caratelli ldquoPlastic-based supershaped dielectric resonator antennas for wide-bandapplicationsrdquo IEEE Transactions on Antennas and Propagationvol 59 no 12 pp 4820ndash4825 2011
[16] C A Balanis AntennaTheory Analysis and Design JohnWileyand Sons Hoboken NJ USA 3rd edition 2005
[17] P Bia D Caratelli L Mescia and J Gielis ldquoElectromag-netic characterization of supershaped lens antennas for high-frequency applicationsrdquo in Proceedings of the 43rd EuropeanMicrowave Conference (EuMC rsquo13) pp 1679ndash1682 NurembergGermany October 2013
[18] P Bia D Caratelli L Mescia and J Gielis ldquoAnalysis and syn-thesis of supershaped dielectric lens antennasrdquo IETMicrowavesAntennas amp Propagation vol 9 no 14 pp 1497ndash1504 2015
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of