Design and characterization of half Maxwell fish-eye lens antennas in millimeter waves
ABSTRACT This paper presents the performance of multilayered half Maxwell fish-eye (HMFE) lenses fed by aperture-coupled microstrip patch antennas. Manufacturing techniques are reviewed and the shell technique is retained. Many lens configurations are investigated and compared using a full-wave electromagnetic software at 50 GHz. We report the effects of the number of shells, diameter of the lens, and distance between the primary source and the lens on the input impedance, broadside directivity, and aperture efficiency. Thus, we show that aperture efficiencies up to 95% can be obtained for a one-wavelength-diameter lens with only three shells, justifying the interest in such lenses. An analytical optimization method is also proposed and detailed to choose the thickness and permittivity of a three-shell HMFE lens to approach the radial permittivity law as well as possible. Simulations of lens antennas whose shell characteristics are determined by various ways show that the optimized lens is the one that provides the highest broadside directivities. Finally, measurements done with a three-shell four-wavelength diameter lens fed by a 2 times 2 patch antenna array show the validity of these simulations. To our knowledge, this represents the first layered HMFE lens carried out in the millimeter-wave frequency range
- [show abstract] [hide abstract]
ABSTRACT: A new approach to wide scan-angle antennas at millimeter-wave frequencies is introduced with special focus on ease of manufacturing and reliability. The system is composed of planar feed antennas (tapered-slot antennas), which are positioned around a homogeneous spherical Teflon lens. Beam scanning can be achieved by switching between the antenna elements. The spherical-lens system is analyzed through a combined ray-optics/diffraction method. It is found that a maximum efficiency of 50%-55% can be achieved using Teflon, Rexolite, or quartz lenses. The efficiency includes taper, spillover, and reflection loss. Calculations also indicate that the maximum lens diameter is 30-40 λ<sub>0</sub>, which results in a maximum directivity of 39.5-42 dB. Measurements done on a single-element feed and a 5-cm Teflon lens agree very well with theory and result in a 3-dB beamwidth of 5.5° and better than -20-dB sidelobe levels at 77 GHz. Absolute gain measurements show a system efficiency of 46%-48% (including dielectric loss). A 23- and 33-element antenna array with a scan angle of ±90° and a -3.5- and -6-dB crossover, respectively, in the far-field patterns was also demonstrated. The 23-element array resulted in virtually no gain loss over the entire 90° scan angle. This represents, to our knowledge, the first wide scan-angle antenna at millimeter-wave frequenciesIEEE Transactions on Microwave Theory and Techniques 10/2002; · 2.23 Impact Factor
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ABSTRACT: Single- and multiple-beam circularly polarized ellipsoidal substrate lenses suitable for millimeter-wave wireless communications have been designed, implemented, and experimentally characterized at 30 GHz. The lenses are made out of low-cost low-permittivity Rexolite material. The single-beam lens achieves a directivity of 25.9 dB, a front-to-back ratio of 30 dB, and an axial ratio of 0.5 dB is maintained within the main lobe. The measured impedance bandwidth is 12.5% within a SWR⩽1.8:1. The single-beam antenna is well suited for broad-band wireless point-to-point links. On the other hand, the multiple-beam lens launches 31 beams with a minimum 3-dB overlapping level among adjacent beams. The coverage of the lens antenna system has been optimized through the utilization of a hexagonal patch arrangement leading to a scan coverage of 45.4° with a maximum loss in directivity of 1.8 dB due to multiple reflections. The multiple-beam lens antenna is suitable for indoor point-to-multipoint wireless communications such as a broad-band local area network or as a switched beam smart antenna. During the proposed design process, some fundamental issues pertaining to substrate lens antennas are discussed and clarified. This includes the depolarization properties of the lenses, the effect of multiple internal reflections on the far-field patterns and the directivity, the nature of the far-field patterns, the estimation of the lens system F/B ratio, and the off-axis characteristics of ellipsoidal lensesIEEE Transactions on Microwave Theory and Techniques 04/2001; · 2.23 Impact Factor
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ABSTRACT: Dielectric lens antennas can be designed to produce highly shaped beams that significantly improve the system performance in emerging wireless indoor millimeter-wave systems. A lens configuration is analyzed in this paper that produces a circularly symmetric cell with uniform spatial power distribution, fairly sharp boundaries, and scalable cell radius. The last characteristic is used to control the reflections at sidewalls. A hemispherical coverage lens antenna is designed for the mobile terminal (MT) to ensure relatively free movement. The impact of these antennas is analyzed in terms of cell coverage and channel time dispersion, considering the effect of cell radius scaling, and MT antenna tilting. Measurements and simulations show that the proposed lens antennas outperform common solutions based on pyramidal horns or biconicsIEEE Transactions on Microwave Theory and Techniques 07/1999; · 2.23 Impact Factor
2292IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
Design and Characterization of Half Maxwell
Fish-Eye Lens Antennas in Millimeter Waves
Benjamin Fuchs, Olivier Lafond, Sébastien Rondineau, Member, IEEE, and Mohamed Himdi
Abstract—This paper presents the performance of multilayered
half Maxwell fish-eye (HMFE) lenses fed by aperture-coupled mi-
crostrip patch antennas. Manufacturing techniques are reviewed
and the shell technique is retained. Many lens configurations are
investigated and compared using a full-wave electromagnetic soft-
eter of the lens, and distance between the primary source and the
ficiency. Thus, we show that aperture efficiencies up to 95% can be
justifying the interest in such lenses. An analytical optimization
method is also proposed and detailed to choose the thickness and
shell characteristics are determined by various ways show that the
optimized lens is the one that provides the highest broadside direc-
tivities. Finally, measurements done with a three-shell four-wave-
length diameter lens fed by a 2
validity of these simulations. To our knowledge, this represents the
first layered HMFE lens carried out in the millimeter-wave fre-
2 patch antenna array show the
Index Terms—High directivity, lens antennas, millimeter-wave
have increased significantly in the last decades. This has lead
to renewed interest in a category of dielectric antennas: lens
antennas. At millimeter-wave frequencies, dielectric lens an-
tennas exhibit reduced weight and size as compared to con-
ventional antennas. These properties are very attractive for em-
lites, and wireless communication systems , . Although
homogeneous lenses are the most widespread structure for mil-
limeter-wave applications, lens antennas can be built from ho-
mogeneous or inhomogeneous materials. This paper focuses
on an inhomogeneous lens antenna: the half Maxwell fish-eye
(HMFE) lens antenna.
UE TO the congestion of the frequency spectrum, the op-
erating frequencies of wireless communication systems
Manuscript received October 7, 2005; revised February 23, 2006. This work
was supported by the Institute of Electronics and Telecommunications of
B. Fuchs, O. Lafond, and M. Himdi are with the Institute of Electronics
and Telecommunications of Rennes, Unité Mixte de Recherche, Center Na-
tional de la Recherche Scientifique 6164, University of Rennes I, Rennes Cedex
35042, France (e-mail: firstname.lastname@example.org; olivier.lafond@univ-
University of Colorado at Boulder, Boulder CO 80309-0425 USA (e-mail: se-
Digital Object Identifier 10.1109/TMTT.2006.875255
Fig. 1. Index distribution along the normalized lens radius in the case of:
Maxwell fish-eye ???, Eaton ? ?, Eaton–Lippman ???, and Luneburg ???.
Among the inhomogeneous lenses, we can distinguish be-
tween the multimaterial lens and the gradient-index lens. For
multimaterial lenses, the most popular structure is the Fresnel
zone plate lens , which is constructed with stacked parallel-
plate waveguides of various lengths. In contrast, gradient index
lenses are spherical or hemispherical lenses whose index of re-
fraction varies from the center to the surface of the lens ac-
cording to a radial refractive index. The most famous laws are
shown in Fig. 1. They are the Luneburg , Maxwell fish-eye
, Eaton , and Eaton-Lippman lenses . The original de-
signs of these lenses are based on geometrical optics and, more
specifically, Fermat’s principle . Extensive studies were con-
ducted from the 1950s to the 1970s to understand the electro-
riety of modifications and generalizations in the design –.
alyze continuous ,  and multilayered lenses, e.g., mode-
matching techniques –. However, these methods do not
account for the metallic part of the source.
Therefore, to be as close as possible to the real lens antenna,
the commercial software CST Microwave Studio was used to
of the HMFE lens (Section IV). This code, a finite integration
technique, is based on a discretized solution of the integral for-
mulation of Maxwell’s equations.
To the best of our knowledge, multilayered HMFE lens an-
tennashavenot been previouslyinvestigated.Sinceour aim was
to realize a prototype, we focused our attention on the ease of
0018-9480/$20.00 © 2006 IEEE
FUCHS et al.: DESIGN AND CHARACTERIZATION OF HMFE LENS ANTENNAS IN MILLIMETER WAVES2293
Fig. 2. Focusing properties shown with an unit incoming plane wave of an HMFE lens seen from: (a) geometrical optics ray tracing and (b) electric field computed
by CST Microwave Studio. The electromagnetic simulation shows a blurred focus region instead of a perfect focus point, as predicted by geometrical optics.
Section II shows the theoretical properties of the HMFE lens
and discusses different manufacturing options. This leads to a
hemispherical layered lens for which the permittivity and thick-
ness of each shell are investigated in Section III. Overall per-
by simulations and measurements are presented in Section IV.
Conclusions and perspectives are then drawn in Section V.
II. THEORETICAL PROPERTIES AND MANUFACTURING
TECHNIQUES OF THE HMFE LENSES
Although the optical and electromagnetic properties of ideal
the fabrication of devices, having smoothly varying dielectric
constants, has proven to be quite difficult. Therefore, we briefly
on how to manufacture such a lens.
A. Theoretical Properties of Maxwell Fish-Eye Lenses
Through a Maxwell fish-eye lens, the energy of a point
source, placed at one side of the lens, converges into a focus
point on the diametrically opposite side of the lens. Due to the
symmetry of the structure, a spherical wave at the surface of
the lens is converted into a local plane wave at the center of
the lens and reemerges as a spherical wave at the surface on
the opposite side. Thus, the radiation pattern of a point source
through the Maxwell fish-eye lens is nearly omni-directional
and a point source on the HMFE lens is highly directive.
Geometrical optics predicts that the HMFE lens transforms a
rays incident upon the flat side of the HMFE lens is focused on
a point as represented in Fig. 2(a). To illustrate, Fig. 2(b) shows
the magnitude of the electric field over the
of a 10-
-diameter HMFE lens illuminated by a plane wave.
Greenwood and Jian-Ming  present the same lens excited by
The behavior of the lens can be adjusted by modifying the
index profile. The Maxwell fish-eye lens can focus from one
arbitrary point source to another, as shown in  and  for
the Luneburg lens. Therefore, the HMFE lens can have a focal
point outside the lens on the axis of revolution. This can be very
useful because a point source on the surface of a lens is very
difficult to obtain.
B. Lens Manufacturing
Fabricating spherical gradient index lenses poses several
practical limitations such as obtaining the desired variable
dielectric-constant characteristics. As we have not found any
prototype of the Maxwell fish-eye lens in the literature, we
will briefly review the primary methods for manufacturing
Luneburg lenses that can be applied to the HMFE.
The most common method is the shell technique, also known
as the onion model method. The lens is fabricated from a fi-
nite number of concentric homogeneous dielectric shells .
The main drawback is the difficulty of maintaining the dimen-
sional accuracy of the shells and the concentricity of the layers.
This can produce inter-shell air gaps that alter the performances
of the lens. To avoid this problem of curvature, Zimmerman
et al.  proposed the tapered holes approach. It consists of
drilling radial holes in a homogeneous dielectric sphere in such
a way that their radii extend from the center to the surface of
the lens to alter its dielectric constant. However, this manufac-
turing process requires a tool that can drill in three dimensions
and make tapered holes. This has led to the slice technique .
The lens is approximated by a pile of homogeneous dielectric
coaxial cylinders and the gradient index is obtained by varying
the hole density. As a result, a large number of different pieces
must be machined, which results in a long manufacturing time.
A steeper gradient requires a smaller hole radii, which increases
the number of holes and slices. Consequently, the HMFE distri-
bution, whose index gradient is steeper than the Luneburg dis-
tribution, would require higher accuracy and increased fabrica-
tion time. This makes the slice technique less appropriate than
the concentric shells to build an HMFE lens.
In view of the methods employed for the Luneburg lens, the
shell technique  is the most appropriate for fabricating the
HMFE lens. With this technique, we add concentric homoge-
neous layers of decreasing dielectric constant, which approxi-
mate the desired radially varying dielectric constant. Accord-
ingly, the lens is called a discrete HMFE lens.
III. OPTIMIZED DISCRETIZATION TECHNIQUES
The discrete HMFE lens is constructed as a multishell hemi-
spherical lens. Some parameters, namely, the number of shells,
thickness, and permittivity of each shell, need to be chosen be-
fore manufacturing the lens.
2294 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
Fig. 3. (a) Exploded three-dimensional (3-D) view of a three-shell HMFE lens. (b) Top view of a quarter of a lens with the six parameters to be found.
A. Number of Shells
In general, a higher number of shells more closely approx-
imates the ideal lens and, therefore, performs better. Thus, to
attain a given performance, there exists a minimum number of
shells for a lens with a given size and source.
As we focus our attention on the ease of manufacturing, we
choose to use only three shells for the lens [see Fig. 3(a)]. We
will prove in Section IV-B why this choice is a good one.
B. Choice of Layer Thickness and Relative Permittivity of
Instead of varying from two to one, as in the Luneburg case,
the relative permittivity ranges from four at the center to one at
thesurfacefor theMaxwellfish-eyedistribution.It is,therefore,
even more important to carefully choose the thickness and the
relative permittivity value of each shell.
Various choices for these parameters have already been pro-
posed for the discrete Luneburg lenses, but no adequate justifi-
cations have been given.Peeler and Coleman  used an equal
dielectric constant step and chose step positions at the radii of
chose the radius of each shell such that the projected area of the
shells is equal. More recently, nonuniform Luneburg lens an-
tennas have been synthesized by genetic algorithms .
of air, which will be taken into account to approximate the law.
We denote the outer radius of this pseudoshell with the variable
. The geometry of the problem is presented in Fig. 3(b). The
is the outer radius of the th shell. Note that
The number of degrees of freedom is then six, namely,
and(i.e., the relative permittivity and
normalized outer radii of the three real shells). We propose to
minimize the following criterion with respect to these param-
resents the reconstructed relative permittivity that will be opti-
mized. Due to the hemispherical symmetry of the lens,
is a piecewise constant function of the radial position ,
can be rewritten
, where– ,in (1)
, namely, and. For
to the standard least squares approach. For
, we penalize the absolute
, one defines
, i.e., over each
interval (shell), we minimize the maximum of the discrepancy.
In our case,
is the Maxwell fish-eye law and equals 
To simplify the presentation, we use the normalized radii
ranges from 4 to 1. To perform the optimization, we di-
vide the unknowns into
We choose a reasonable initial point for the radii
and take the minimum of
to get. We then minimize
, etc., i.e., we proceed by minimizing
with respect to
and until a stopping criterion is satisfied. For
the three values of , the details of the analytical calculations
are given in the Appendix. Since the relative permittivity of ma-
terials is usually given with a precision on the order of 1%, we
perform the optimization up to three digits and fix the stopping
We observe that the algorithm converges rapidly and the op-
monotonically decreasing. Therefore, the optimization method
can be applied to any given law provided it is monotonic and
generalized to any number of shells.
. Whilevaries from 0 to 1,
with respect to
IV. PERFORMANCE OF THE DISCRETE HMFE LENS
Here, we first investigate the performance of various HMFE
lenses fed by an aperture-coupled microstrip (ACM) patch an-
tenna. We then compare methods for choosing the lens param-
eters. Finally, experimental results of a three-shell HMFE lens
FUCHS et al.: DESIGN AND CHARACTERIZATION OF HMFE LENS ANTENNAS IN MILLIMETER WAVES2295
Fig. 4. (a) Cross-sectional view of the ?-shell HMFE lens antenna. Top view of the primary sources: (b) an ACM single and (c) 2 ? 2 patch antenna array.
CHARACTERISTICS OF THE PRIMARY SOURCES
fed by an ACM 2
pared to simulations.
2 patch antenna array are given and com-
A. Antenna Geometry
The geometry of the lens antennas and primary sources is
relative permittivity is
-shell lens. The diameter of the lens is labeled
notes the distance between the source and lens. The operating
frequency of the two primary sources is 50 GHz. Most of the
simulations (Sections IV-B and IV-C) were done with the ACM
single-patch antenna to minimize the computing time. As we
wanted to have a more significant directivity to reduce diffrac-
tion effects at the edges, an ACM 2
was realized, measured, and compared to simulations. The di-
mensions and substrate characteristics of both sources are given
in Table I. In order to investigate the properties and influence of
the various lens parameters, we consider a uniform HMFE lens.
This means that each shell has the same thickness and its per-
mittivity is such that
is the outer radius of the
2 patch antenna array
B. Study of the Uniform HMFE Lens
To understand the influence of the number of shells consti-
the variations of the reflection coefficient
the number of shells. Increasing the number of shells improves
the match and smooths the change in permittivity between the
different layers. Note that, with only three shells, the matching
is already good ( 20 dB). Thus, for these lens antennas, the de-
sign of the source can be done independently of the lens. Varia-
tions of the maximum directivity
and the aperture efficiency
a function of the number of shells. The aperture efficiency
is defined by, where
a constant field circular aperture of the same diameter
Note that, for the patch alone, we have
lens diameter, increasing the number of shells above three does
not result in a significant increase in directivity.
Let us now see the influence of the diameter of the lens in
Fig. 6 by considering the following configuration: an ACM
single-patch source, a three-shell uniform HMFE lens, and
. Increasing the diameter of the lens enhances the
directivity of the lens antenna. However, this leads to a larger
radiating aperture and, thus, is accompanied by a reduction of
While retaining a uniform lens and
between the primary source and the lens to im-
prove the performance of the lens antenna. Fig. 7 shows the in-
by an ACM single patch. The relative variation of the broadside
directivityis defined as
By sorting the three lens antennas by increasing
the three-, five-, and ten-shell lens fed by the ACM patch. Thus,
the patch with the three-shell lens is the configuration that is
the least sensitive, in terms of directivity, to the distance . The
influence of this distance, although difficult to quantify, can be
explained qualitatively in the following way: the closer to the
which shows the magnitude of the electric field over the
-plane of a three-shell -diameter HMFE lens illuminated
by a plane wave. We notice that the focal zone is relatively wide
and blurred. Moreover, simulations performed with a dipole in-
and. Fig. 5(a) shows
as a function of
in the broadside direction
are represented in Fig. 5(b) as
is the directivity of
dB. For a given
, one can exploit
, we get
2296IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
Fig. 5. Influence of the number of shell: (a) on the return loss and (b) on the broadside directivity ? .Directivity of the primary source.
Fig.6. Variationofthebroadsidedirectivity? andtheapertureefficiency?
of the three-shell uniform HMFE lens as a function of the lens diameter ?.
stead of a patch show that
theelectricalsizeof theantenna decreases,thelensantennaper-
formance becomes more sensitive to the distance .
is then even more important. As
C. Comparison Between Methods for Choosing the
It is important to know how to choose the lens parameters
in order to maximize the directivity of the lens antenna. For
that purpose, we consider three-shell lenses whose permittivity
and shell thickness are chosen by five different methods, which
are: three optimized lenses (with
lens; and the equal area lens, for which
these lenses are detailed in Table II. The broadside directivities
of these five three-shell 4-
-diameter lenses, fed by the ACM
patch, are investigated as a function of the distance , as shown
Its main effect is that the thickness of the air shell predicted by
optimization is different from the distance .
We notice that the lens optimized with
highest directivities—a result that validates the choice of this
and); the uniform
. The parameters of
leads to the
Fig. 7. Influence of the distance ? on the broadside directivity ? for three
uniform HMFE lens fed by an ACM patch: a three-shell lens ? ?, five-shell
lens ???, and ten-shell lens ? ?.
Fig. 8. Electric field distribution, computed by CST Microwave Studio, in the
neighborhood of a ?? -diameter three-shell uniform HMFE lens excited by
a unit incoming plane wave. The electromagnetic simulation shows a blurred
focus region at the bottom of the lens.
approximation criterion (see Section III-B for the definition).
The radii and permittivities are chosen such that, everywhere in
the lens, the difference between the approximated permittivity
FUCHS et al.: DESIGN AND CHARACTERIZATION OF HMFE LENS ANTENNAS IN MILLIMETER WAVES2297
CHARACTERISTICS OF THE SIMULATED LENSES
Fig. 9. Comparison of the directivity ? as a function of the distance ? for
five three-shell 4-? diameter lenses: the optimized lens with ? ? ? ? ?, with
? ? ? ???, with ? ? ? ? ?, the uniform lens ???0, and the equal area lens
and the theoretical permittivity is as small as possible. This is
the maximum discrepancy over the entire domain is minimized.
in Appendix I.
D. Experimental Results
The primary source is the 2
printed on a 256- m-thick RT 6006 Duroid substratewhose rel-
ative permittivity has been characterized to 7.0 at 50 GHz. A
low loss and low dielectric foam support (Eccostock SH-2 with
the primary source. The lens has an outer diameter of 24 mm,
at 50 GHz. The three shells of the lens have relative
andmm, respectively. They are made by Emerson
antenna is shown in Fig. 10. The primary source is located at
from the lens in order to maximize the
directivity of the lens antenna.
Fig. 11 compares the simulated and measured far-field radia-
tion patterns, at 48.5 GHz, of the primary source alone and with
is maximal and reaches 16.4 dB. The total efficiency of the
lens antenna is then
has been previously defined, and
2 patch array in Fig. 4(c)
andwith boundaries at
). A top view of this lens
. The aperture efficiency
is the efficiency due to the
Fig. 10. Top view of the three-shell HMFE lens and its foam support.
For a computed directivity
ciency of the system is
The measurements and simulations agree quite well, which
validates the simulations previously shown.The main beam and
peak positions are well predicted. The differences in the side-
lobe levels are due to the diffraction effects of the ground plane
and the V-connector. This is particularly visible in the
pattern of the primary source without the lens. The disagree-
ments concerning the main beam width may be due to an error
in the positioning of the source since the device used to fix the
lens does not make it possible to have a precision of
than 0.1 mm.
dB, the aperture effi-
and the efficiency due to
This paper presented an investigation on the HMFE lens in
the millimeter-wave domain. Manufacturing techniques were
reviewed, yielding the shell technique as a viable option. An
optimization method to choose the thickness and permittivity of
each shell was described for a three-layered HMFE lens. This
method depends upon a parameter we have denoted , and it is
The highest directivity is obtained for
sponds to a minmax optimization. Many configurations of mul-
tishell HMFE lenses fed by an ACM patch have been computed
and compared showing that the limitation to three shells is a
good compromise between ease of manufacturing and perfor-
mance in terms of directivity.
2patch array.Comparison betweenthemeasuredand
computed patterns at 48.5 GHz validates the simulation results
and the interest in such a lens. Moreover, the results regarding
the input impedance of the lens antenna show that the presence
can be made by considering the primary source alone. Since the
can be applied.
The directivity and aperture efficiency for the HMFE lens an-
tenna make it especially suitable for embedded communication
systems in millimeter-wave frequencies. For that purpose, we
, which corre-
2298IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 6, JUNE 2006
Fig. 11. Comparison between simulated (---) and measured [co-pol (—) and cross-pol ?????] both ?- and ?-plane radiated far-field pattern at 48.7 GHz for:
(a) and (b) the primary source alone and (c) and (d) with the three-shell HMFE lens, respectively.
are currently investigating a 77-GHz HMFE lens antenna for
automotive radar applications.
II. ANALYTICAL AND IMPLEMENTATION CONSIDERATIONS OF
THE OPTIMIZATION ALGORITHM
Detailed here is the optimization algorithm presented in Sec-
tion III-B. The following expression has to be minimized:
both the layer thickness and relative permittivity of each lens
shell, as shown in Fig. 12. The notations are defined in Fig. 13.
Note that, in the sequel, to perform the integrations involving
the theoretical relative permittivity
order 3 polynomial
found by the least squares approach.
We thus replace
This allows analytical integration.
and. The implemented algorithm optimizes
, we approximate it by an
Fig. 12. Chart of the optimization algorithm.
FUCHS et al.: DESIGN AND CHARACTERIZATION OF HMFE LENS ANTENNAS IN MILLIMETER WAVES 2299
Fig. 13. Representation of the theoretical (—) and reconstructed (---) permit-
tivity laws and notations used for the normalized radial dimensions and relative
For, we have
Let us introduce the intermediate variables
. Such an
is a monotonic function. Therefore, the absolute values
can be dropped, which eases the calculations of the derivatives.
We then obtain
always exists and is unique since
Step 1, the optimization with respect to
and it follows that
. Step 2, the optimization with respect to
, leads to
, from which 1 deduces
is monotonically decreasing. As the
has no analytical inverse, the evaluation of the
’s requires the implementation of a local and trivial search
algorithm. One then alternates steps 1 and 2 until the stopping
criterion is satisfied.
, we have
Nulling the derivatives with respect to the
step 2 local optimization with respect to the
’s, step 1
. While the
’s leads to
. We proceed
to obtain the’s.
, one observes that sinceis mono-
tonic, the maximum of
obtained at one of the two boundaries of the interval. For
fixed ’s, the optimum in step 1 is then simply attained when
over each interval is
, i.e., when the discrepancy is
the same at both ends of each interval. Thus, the first iteration
consists of the optimization of (9) with respect to the
,. The second step then
This leads to an easy-to-solve linear system that successively
, and finally,
deduces the optimal
’s. The optimization in the case
can be done analytically and needs no iterative algorithm.
from which one
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Benjamin Fuchs was born May 28, 1981. He
received the Electronics Engineering degree and
French DEA(Masters) degree in electronicsfrom the
Institut National des Sciences Appliquées (INSA),
Rennes, France, in 2004, and is currently working
toward the Ph.D. degree in signal processing and
telecommunications at the Institut d’Electronique
et de Télécommunications de Rennes (IETR),
University of Rennes 1, Rennes, France.
His research interests are millimeter-wave fo-
cusing and multibeam devices. His focus is on
Olivier Lafond received the French DEA (Masters)
degree in radar and telecommunications from the
University of Rennes, Rennes, France, in 1996,
and the Ph.D. degree in signal processing and
telecommunications from the University of Rennes
1, Rennes, France, in 2000.
Since October 2002, he has been an Associate
Professor with the Institute of Electronics and
Telecommunications of Rennes (IETR), University
of Rennes 1. His research interests are passive
and active millimeter-wave antennas and circuits,
multibeam antennas, inhomogeneous lenses, and substrate characterization
techniques for millimeter-wave applications.
Sébastien Rondineau (M’04) was born in Paim-
boeuf, France, in 1975. He received the Diplôme
d’Ingénieur en Informatique et télécommunications
(a postgraduate degree in signal processing and
telecommunications) and Ph.D. degree from the
University of Rennes 1, France, Rennes, France, in
1999 and 2002, respectively.
trical and Computer Engineering Department, Uni-
versity of Colorado at Boulder. His research interests
include the method of analytical regularization in computational electromag-
netics,mode matching,conformal mapping, microscaleinterconnects,propaga-
discrete lens arrays and antennas.
processingand telecommunications from the Univer-
sity of Rennes 1, Rennes, France, in 1990.
Since 2003, he has been a Professor with the
University of Rennes 1, and is currently the Head
of the High Frequency and Antenna Department,
Institut d’Electronique et Télécommunications de
Rennes (IETR), Unité Mixte de Recherche, Center
National de la Recherche Scientifique. He has
authored or coauthored 36 journal papers and over
120 papers in conference proceedings. He has also
authored/coauthored two book chapters. He holds eight patents in the area of
antennas. His research activities concern passive and active millimeter-wave
antennas. His research interests also include theoretical and applied computa-
tional electromagnetics, development of new architectures of printed antenna
arrays, and new three-dimensional (3-D) antenna technologies.
Prof. Himdi was the recipient of the 1992 International Symposium on An-
tennas and Propagation (ISAP) Conference Young Researcher Scientist Fellow-
ship (Japan) and a 1995 award presented by the International Union of Radio
Scientists (Russia). He was Laureat of the Second National Competition for the
and Education, France).