Proceedings of the 43rd European Microwave Conference
Electromagnetic Characterization of Supershaped Lens Antennas for High-Frequency Applications P. Bia#1, D. Caratelli*,◊2, L. Mescia#3, J. Gielis+4 #
Politecnico di Bari Via E. Orabona 4, 70125 Bari, Italy 1
[email protected],
[email protected] *
Delft University of Technology Mekelweg 4, 2628 CD Delft, the Netherlands ◊ Antenna Company B.V. Grebbeweg 111, 3911 AV Rhenen, the Netherlands 2
[email protected] +
University of Antwerp Groenenborgerlaan 171, 2020 Antwerp, Belgium 4
[email protected]
Abstract—A novel class of supershaped dielectric lens antennas, whose geometry is described by the three-dimensional extension of Gielis’ formula, is introduced and analyzed. To this end, a hybrid approach based on geometrical and physical optics is adopted in order to properly model the multiple wave reflections occurring within the lens and the relevant impact on the radiation properties of the antenna under analysis. The developed modeling procedure has been validated by comparison with numerical results already reported in the scientific literature and, afterwards, applied to the electromagnetic characterization of a Gielis lens antenna with shaped radiation pattern. Keywords—Lens antennas, Gielis formula, high-frequency techniques.
I.
INTRODUCTION
Dielectric lens antennas are widely used in a number of applications such as radar [1], millimeter wave imaging [2], radio-astronomy [3], as well as broadband wireless communications at high frequencies [4]. The attractive feature of this class of antennas primarily consists in the beam collimating/shaping capability, combined with both mechanical and thermal stability, which eases the integration in different electronic circuits. In the scientific literature, a great deal of attention has been devoted to dielectric lenses with canonical geometries, elliptical, spherical, or hyperbolical [5]-[6], optimized in order to enhance the directivity of the antenna and eventually the Gaussicity of the radiated beam. Few scientific studies available in the literature deal with lens antennas featuring a more complex geometry [7]-[8]. On the other hand, the effect of multiple internal wave reflections has been investigated only in lens antennas with classical shape, such as the spherical [9], or the elliptical ones [10].
978-2-87487-031-6 © 2013 EuMA
The goal of this research is to present a preliminary study of a new class of supershaped dielectric lens antennas whose geometry is described by the so-called Gielis’ formula [11]. This formula, which generalizes the ellipse's polar equation, allows the modeling of an extremely wide range of natural objects (plants, stems, starfish, shells, flowers, and more) alongside man-made structures, in a simple and analytical way by means of a reduced number of parameters. This, in turn, translates into the possibility of automatically reshaping the lens profile so that any automated optimization procedure could be conveniently adopted in order to identify the geometrical parameters yielding optimal antenna performance. In the presented contribution, the radiation properties of the mentioned antennas are investigated by means of a dedicated high-frequency technique based on optical ray approximation [12]. In particular, geometrical optics (GO) is adopted to analyze the electromagnetic field propagation within the lens region. In doing so, the contribution of the multiple internal reflections is properly taken into account, so enhancing the accuracy of the modeling procedure especially where dielectric materials with relatively large permittivity are considered in the design. As a matter of fact, in this case, the common hypothesis that the energy content relevant to higher order reflected rays can be neglected is not applicable. Finally, by virtue of the equivalence principle, the electromagnetic field outside the lens can be evaluated by radiation in free space of the equivalent electric and magnetic current distributions along the interface with the air region. In the developed methodology, these currents are determined by application of the local Fresnel transmission coefficients along the surface of the lens to the GO fields in accordance with the physical optics (PO) method [12]. In this way, differently shaped Gielis’ lens antennas may be characterized in a reasonably accurate way, while reducing both computational resources and design time.
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II.
ANTENNA MODELING APPROACH
As shown in Fig. 1, the general antenna structure considered in this study consists of an electrically large dielectric lens placed at the center of a perfectly conducting metal disk, with radius d p , acting as a ground plane and, at the same time, a shield useful to reduce the backscattered radiation. The lens is illuminated by the electromagnetic field excited by a radio source, such as an open-ended waveguide, horn antenna, or coaxial probe, which is modeled by means of the relevant far-field pattern [13].
G m nˆ × kˆim m , E i ⊥ = Ei ⋅ m nˆ × kˆi
(3)
Gm Gm ˆ × kˆim m n . E i & = E i − Ei ⊥ m nˆ × kˆi
(4)
G The incident field Eim ( P ) at the general point P is directly computed by using the far-field pattern of the source if G m = 1 . On the other hand, for m > 1 , Eim ( P ) is derived starting from the m − th reflected wave contribution as: G G Eim ( P ) = Erm ( Pm ) e jki d , (5)
with d denoting the Euclidean distance between the observation point P and the point Pm at which the reflection G takes place. The reflected field Eim ( Pm ) appearing in (5) is given by: G G G j∠E m Erm ( Pm ) = Erm& e r &
( nˆ × kˆ ) × kˆ ( nˆ × kˆ ) × kˆ m r
m r
m r
m r
+ Erm⊥
nˆ × kˆrm , nˆ × kˆ m
(6)
r
where the parallel and orthogonal components with respect to G the plane of incidence, Erm& and Erm⊥ , are computed by multiplication of the corresponding components, Eim& and Eim⊥ ,
Fig. 1. Geometry of a dielectric lens antenna.
Under the mentioned assumptions, the electric field distribution transmitted outside the lens region can be conveniently evaluated as: G G Et = Etm , (1)
∑ m
with: G G G j∠E m Etm = Etm& e t&
( nˆ × kˆ ) × kˆ ( nˆ × kˆ ) × kˆ m t
m t
m t
m t
+ Etm⊥
nˆ × kˆtm nˆ × kˆ m
(2)
t
denoting the electric field contribution pertaining to the internal reflection process of order m . In (2), nˆ denotes the normal to G G the surface of the lens, kˆ m = k m k m is the normalized wave t
t
of the incident field at the point Pm with the proper Fresnel G G reflection coefficients. In (6), kˆim = kim kim is the normalized Gm wave vector of the incident field with k i = 2π nd λ0 , nd being the refractive index in the dielectric material forming the lens. Once the GO field has been evaluated, the equivalent G G electric J S and magnetic M S current densities along the surface of the lens can be determined in a straightforward manner. In this way, according to the PO method, the electromagnetic field radiated by the antenna at the observation point PFF ≡ ( rFF , θ FF , φ FF ) can be readily computed by means of the integral expression [7]: − jk r G e E FF ( PFF ) = j 2λ0 rFF 0 FF
t
propagation vector of the m-th reflected beam transmitted out Gm of the lens with kt = 2π λ0 , λ0 being the operating G wavelength in the vacuum. In particular, Etm& and Etm⊥ are the parallel and orthogonal components, respectively, of the transmitted wave contribution due to the m − th reflected beam. These field quantities are determined by multiplying the G G components Eim& , Eim⊥ of the internal field Eim impinging on the surface of the lens with the proper Fresnel transmission G coefficients. The evaluation of Eim& , Eim⊥ is, in turn, carried out by means of the following relations [7]:
G
∫ [η J ( P ) × uˆ 0
S
G G jk r ⋅uˆ − M S ( P )] × uˆ0 e dS , 0
0
0
S
(7) where
uˆ0
is the unit vector, at the general point
P ≡ ( r , θ , φ ) on the surface, pointing to PFF .
In the proposed study, the geometry of the lens is assumed to be described by the three-dimensional extension of the socalled superformula, introduced by Gielis in [11] in order to describe complex natural and abstract shapes in a simple and analytical way. To this end, the following expression is used:
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G r (θ (θ G , φG ) , φ (θ G , φG ) ) = G = r (θ (θ G , φG ) , φ (θ G , φG ) ) r =
(8)
2 2 2 G z +y +x r
with:
⎧ x = x (θ G , φG ) = R (θ G ) cos (θ G ) R (φG ) cos (φG ) ⎪ ⎨ y = y (θ G , φG ) = R (θ G ) sin (θ G ) R (φG ) cos (φG ) , ⎪ z = z (θ , φ ) = R (φ ) sin (φ ) G G G G ⎩
Fig. 2, the lens has a diameter dl = 30.5 mm with extension length h = 16.69 mm , and is assumed to be placed on an infinite ground plane and fed by a circular waveguide filled with the same dielectric material, that is Rexolite with refractive index nd = 1.59 .
(9)
and: −
1
⎡ ⎛ m1θ G ⎞ ⎛mθ ⎞ ⎤ sin ⎜ 2 G ⎟ ⎥ ⎢ cos ⎜ 4 ⎟ ⎝ ⎠ + ⎝ 4 ⎠ ⎥ , R (θ G ) = ⎢ a1 a2 ⎢ ⎥ ⎢⎣ ⎥⎦ n1
n2
n ⎡ ⎛ m3φG ⎞ n m4φG ⎞ ⎤ ⎛ sin ⎜ ⎟ ⎥ ⎢ cos ⎜ 4 ⎟ ⎝ ⎠ + ⎝ 4 ⎠ ⎥ R (φG ) = ⎢ a3 a4 ⎢ ⎥ ⎢⎣ ⎥⎦ 3
−
4
b1
(10)
1
Fig. 3. E-plane radiation pattern of the synthesized elliptical lens shown in Fig. 2 as function of the elevation angle.
b2
,
(11)
The normalized radiation pattern excited along the E-plane of the antenna at the working frequency f = 29.5 GHz is shown in Fig. 3. As it appears from this diagram, a substantial
where n1 , n2 , a1 , a2 , m1 , m2 , n3 , n4 , a3 , a4 , m3 , m4 , b1 , b2 are real-valued parameters selected in such a way that the surface of the lens is actually closed and characterized, at any point, by curvature radius larger than the working wavelength in accordance with the GO approximation. dl
h
(a)
dg dp Fig. 2. Three-dimensional view of a synthesized elliptical lens fed by a dielectric-loaded circular waveguide. Antenna characteristics: d g = 5.2 mm , dl = 30.5 mm , h = 16.69 mm , d p → +∞ . The lens is
made out of rexolite with refractive index of 1.59. (b)
III.
NUMERICAL EXAMPLES
In order to validate the developed modeling approach, the electromagnetic characterization of the elliptical lens antenna analyzed in [14] has been carried out. As it can be observed in
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Fig. 4. Geometry (a) and radiation solid (b) of a supershaped lens antenna fed by an open-ended rectangular horn antenna filled with the same material of the dielectric lens, that is Silicon with refractive index nd = 3.42 .
agreement with the numerical results reported in [14] has been obtained, thus confirming the accuracy of the hybrid GO/PO methodology briefly discussed in the previous section.
[2]
[3]
[4]
[5]
[6]
[7]
[8] Fig. 5. H-plane radiation pattern of the supershaped lens antenna shown in Fig. 4.
[9]
By using the illustrated modeling technique, a supershaped lens antenna has been designed in order to achieve a flat-top radiation pattern at frequency f = 10 GHz . The lens, made out of a dielectric material with refractive index of nd = 3.42 , features a maximum radius rmax = 10cm , and is described by the Gielis’ parameters n1 = 1 , n2 = 1 , a1 = 1 , a2 = 1 , m1 = 3 , m2 = 3 , n3 = 1 , n4 = 1 , a3 = 1 , a4 = 1 , m3 = 3 , m4 = 3 , b1 = 2 , b2 = 2 . A rectangular horn antenna filled up with the same material forming the lens is used for feeding the structure. In particular, the input port of the horn has dimensions ain = 6.7 mm and bin = 3mm , whereas the dimensions of the output port are aout = 50mm and bout = 25mm , the total length being l = 300mm .
[10]
[11]
[12] [13] [14]
In particular, the numerical simulations have been performed by using a suitable tube-tracing formulation, and accounting for internal reflections of order up to m = 4 in order to achieve a sufficient accuracy in the modeling. Figure 4 shows the considered lens geometry, as well as the relevant radiation solid which is characterized by a flat-top distribution with −3dB angular aperture of about 90 degrees (see Fig. 5). As discussed in [4], this performance is very attractive in emerging applications of meshed networks for wireless communications at high frequencies. In this context, it becomes essential to shape efficiently the radiation pattern of the antennas integrated in access points and user terminals in order to focus the power in the desired coverage zone, thus improving the quality of the radio link and, at the same time, reducing the electromagnetic interferences with other devices. REFERENCES [1]
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