Design of axis-symmetrical dome antennas with secant-squared beams
G. Godi, R. Sauleau, D. Thouroude
IETR – Institut d’Electronique et de Télécommunications de Rennes, www.ietr.org
UMR CNRS 6164, Université de Rennes 1 – Campus de Beaulieu – Bât. 11D
263, Avenue du Général Leclerc, 35042 Rennes Cedex, France
E-mail: [email protected]
Abstract1— In this paper we describe the design of shaped
dielectric dome antennas radiating a secant-squared beam at
60GHz. The design methodology consists in a binary Genetic
Algorithm combined with an asymptotic method of analysis
based on the Geometrical Optics / Physical Optics technique.
The dome antenna is made in Rexolite and is excited by a
compact pyramidal horn antenna. The use of matching layers
coated on both sides of the dome enables one to reach a 5GHz
band around the central frequency.
I. INTRODUCTION
Dielectric lens antennas have been extensively studied, and
many homogeneous and non-homogeneous configurations
have been proposed [1][2]. Among those, the so-called
Integrated Lens Antenna (ILA) is probably the most popular
one, due to its low cost and high performance. A number of
papers have reported their use for millimeter and submillimeter wave applications, like point-to-point and pointto-multipoint radio links [3][4], imaging systems and quasioptical receivers [5][6], satellite communications and
instrumentation [7][8]. In these works, two main kinds of
lenses are employed, namely the extended hemispherical
lenses and the shaped lenses. In the second case, the shape of
the lens profile is computed so that its radiation pattern
complies with given specifications. The appropriate
configuration for the lens is generally selected according to
the target application (high-gain, multiple-beam or shaped
beam antennas). In any case, the lens is bulky and heavy, and
is usually fed by a planar antenna or an open-ended metallic
waveguide in direct contact with the lens body.
The objective of this paper is to analyze the capabilities of
dome antennas [9][10] for beam shaping applications [11]. In
contrast to standard ILAs, the dome antennas have a much
lower weight and are compatible with the use of 3-D primary
feeds.
This paper is organized as follows. The design tool
developed at IETR for the analysis and optimization of multishell arbitrarily-shaped lenses is described in Section II. Then
a shaped dome antenna is designed at 60GHz in Section III;
both dielectric interfaces of the dome are optimized in order
to radiate an axis-symmetrical secant-squared beam. The role
of the multiple internal reflections is also briefly highlighted.
Conclusions are finally drawn in Section IV.
II. DESCRIPTION OF THE OPTIMIZATION TOOL
One possible approach for the optimization of ILAs is a
two-step method [12]. It first consists in 1) resolving the
inverse problem using GO laws in order to determine the
theoretical lens profile satisfying the far-field specifications,
and then 2) in locally optimizing this shape using a more
accurate
method
of
analysis,
such
as
the
geometrical / physical optics (GO/PO) approach [6][12].
However, in the first step, it is assumed that the lens
dimensions are large compared to the operating wavelength.
If this GO-based algorithm is applied to reduced size lenses
(diameter smaller than a few wavelengths), it cannot be
stated that the shape found with the inverse method is close
enough to the optimal solution so that the local optimization
could be performed efficiently.
To overcome this limitation, we propose to use a singlestep optimization procedure based on a genetic algorithm
(GA) [13] combined with the hybrid GO/PO method.
The flowchart of the design methodology is represented in
Fig. 1. The main input parameters of the optimizer are the
following: 1) design specifications (far-field patterns, power
transfer efficiency), 2) design constraints (central height of
the lens, assembling constraints when dealing with multipleshell ILAs), 3) far-field radiation patterns of the primary
source (in amplitude and phase), and 4) available dielectric
material(s).
The design procedure comprises three main steps:
-
A set of optimum lens shapes is computed using a GA
combined with a method of analysis of 3-D generic
ILAs based on the GO/PO;
-
Then, for further validation purposes, the GO/PO
radiation characteristics of the optimized ILAs (3-D
far-field patterns, axial ratio, etc.) are compared to
those computed with a global electromagnetic
software (in our case, a 3-D home-made FDTD
solver);
-
Finally, the optimized antenna prototype is fabricated
using computer numerically controlled (CNC) milling
facilities for experimental validation.
The GA chosen here is a standard implementation of this
class of optimization methods. The parameters to be
optimized are encoded with a binary representation. The
genetic operators are a tournament-based selection associated
to a double-point crossover of the chromosomes, and a
binary mutation operator. As we do not consider any initial
guess for the lens shape, the initial population of individuals
is a set of random binary chromosomes.
The shape of the lens is represented by a binary string (i.e.
a chromosome). The relation between a physical dome shape
and a chromosome is obtained from the discretization of a
given number of control-points uniformly distributed over
each dome interface. Their shape is reconstructed from these
control-points using a two-dimensional cubic splines
interpolation. This step is of crucial importance because it
represents the link between the real space of possible
solutions and the space of solutions attainable by the
optimizer.
One of the major issues of GA optimization is to get the
most compact chromosome, with the largest feasible solution
space, because most compact the chromosomes are, the
fastest the convergence is. To this end, we implemented a
differential coding [14] of the control-points so as to get a
compact description of the lens shape.
follows:
N −1 M −1
Fitness = Δθ × Δϕ × ∑ ∑ GGO / PO (θ u , ϕ v ) − h(θ u , ϕ v ) ²
where GGO/PO and h denote the far-field radiation pattern of
the ILA under test and the target specifications, respectively.
These patterns are computed for an arbitrary number of
observation points (θu, φv) distributed in space.
More details about the optimization algorithm are given in
Ref. [15].
III. DESIGN OF A DOME ANTENNA
A. Antenna specifications and primary source
The target pattern of the dome antenna is a rotationally
symmetric secant-squared beam at 60GHz. It is represented
in Fig. 2. The primary feed is a TE10-mode distribution
aperture on ground plane. Its radiation pattern is nearly axissymmetrical and has a very small cross-polarization
component (Fig. 3).
Fig. 2. Amplitude template at 60GHz.
Fig. 1. Flowchart of the design methodology.
The far-field properties of each individual of the GA
population are used to assess its performance, in comparison
with the desired specifications. The value of the user-defined
cost function, called Fitness in GA terminology, is defined as
(1)
u =0 v =0
Fig. 3. Radiation pattern of the feed at 60GHz in polar
coordinates. (a) Co-polarization component. (b) Crosspolarization component.
B. Optimized dome antenna
The dome antenna is in Rexolite (εr=2.53, tanδ=6×10-4).
Both interfaces are optimized with the following set of
parameters: 1) the size of the chromosomes and population is
equal to 104 bits and 1024, respectively; 2) the mutation and
crossover probabilities are set to conventional values
recommended in the literature (5% and 90%, respectively).
Figs. 4a and 4b represent 1) the variation of the fitness for
five independent runs of the optimization tool as a function
of number of evaluations, and 2) the best fitness value
obtained for each run. The average value of the fitness
function is very small (<10-4), confirming thereby that the
performance of the synthesized domes is in satisfactory
agreement with the specifications.
The corresponding profiles of the five domes antennas are
given in Fig. 5. Their maximum thickness and external
diameter are in the order of 33mm and 32mm respectively.
(a)
(b)
Fig. 6. 3-D shape, lens profile and radiation patterns in Eplane (solid black line) and H-plane (dashed black line) of
the optimized dome antenna at 60GHz.
(a) Run #2. (b) Run #5.
(a)
(b)
Fig. 4. (a) Convergence of the GA for five independent
runs. (b) Best fitness value for each run.
Fig. 5. Computed lens profiles for five runs of the
optimization tool. The dashed and solid lines represent the
inner and outer dielectric interfaces respectively.
The radiation patterns computed over a 5GHz frequency
band are plotted in Figs. 7 and 8. They are in excellent
agreement with the target template. The moderate side lobe
level obtained in E-plane for grazing incidence (-18dB)
originates from the feed pattern itself (Fig. 3). It could be
reduced significantly using a different feed, such as a doubleslot antenna [6].
Fig. 7. Computed radiation pattern in E-plane.
The 3-D shape and the far-field radiation patterns of the
best individuals obtained from run#2 (fitness=4×10-4) and
run #5 (fitness=0) are given in Figs. 6a and 6b, respectively.
Their radiation performance is in full agreement with the
target specifications. The largest ripples observed in Fig. 6a
in the secant-squared part of the beam explain why the
minimum fitness of run#2 is higher than for run#5 (Fig. 4b).
Fig. 8. Computed radiation pattern in H-plane.
Further numerical simulations with GO/PO and FDTD have
highlighted the considerable effects of the multiple internal
reflections [16]. The use of quarter wavelength matching
layers coated on both sides of the dome enables one to
improve significantly the lens performance. The
corresponding results will be given and discussed during the
conference.
[9]
[10]
IV. CONCLUSIONS
In this preliminary study, we have investigated the
performance of shaped dome antennas for shaped beam
applications. Our design methodology is based in a binary
Genetic Algorithm combined with the Geometrical Optics –
Physical Optics method of analysis.
To illustrate the capabilities of these antennas, both
dielectric boundaries of a rotationally symmetric dome made
in Rexolite have been optimized at 60GHz. The dome
antenna is fed by a compact pyramidal antenna etched in a
metallic plane. The use of anti-reflection coatings enables
one to reduce significantly the influence of multiple internal
reflections.
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