Received October 5, 2020, accepted November 23, 2020, date of publication November 26, 2020,
date of current version December 11, 2020.
Digital Object Identifier 10.1109/ACCESS.2020.3040898
Bessel Beam Generation Using Dielectric Planar
Lenses at Millimeter Frequencies
ÁLVARO F. VAQUERO 1 , (Student Member, IEEE), MARCOS RODRIGUEZ PINO 1 ,
MANUEL ARREBOLA 1 , (Senior Member, IEEE),
SÉRGIO A. MATOS2,3 , (Senior Member, IEEE), JORGE R. COSTA 2,3 , (Senior Member, IEEE),
AND CARLOS A. FERNANDES 2 , (Senior Member, IEEE)
1 Group
of Signal Theory and Communications, Department of Electrical Engineering, University of Oviedo, 33202 Gijón, Spain
de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal
de Ciências e Tecnologias da Informação, Instituto Universitário de Lisboa (ISCTE-IUL), 1649-026 Lisbon, Portugal
2 Instituto
3 Departamento
Corresponding author: Álvaro F. Vaquero (fernandezvalvaro@uniovi.es)
This work was supported in part by the Ministerio de Ciencia, Innovación y Universidades under Project TEC2017-86619-R
(ARTEINE), in part by Gobierno del Principado de Asturias and Fondo Europeo de Desarrollo Regional (FEDER) under Project GRUPINIDI/2018/000191, and in part by the COST (European Cooperation in Science and Technology) under COST Action TD1301, MiMed.
ABSTRACT In this work a dielectric planar lens is proposed to generate a Bessel beam. The lens works at
Ka-band and produces a non-diffraction range within the Fresnel region of the antenna. The methodology
to design the aperture antenna at millimetre or microwave frequencies is presented. It is applied to a
dielectric planar lens made up of cells that shapes the radiated near-field by adjusting the unit cell response.
An approach based on a second order polynomial is proposed to consider the angular dependence of the
phase-shift response of the cell in the designing process. In order to implement the lens physically, two novel
cells, based on rectangular and hexagonal prisms, are proposed, and their performance is compared. The cells
ensure the index dielectric media variation using airgaps to control the overall density of the material. After
fully characterizing the cells, a design is carried out for the two proposed type of cells. The requirement
for the Bessel beam is a depth-of-field of 650 mm at 28 GHz. After evaluating the design in a full-wave
simulation, both prototypes were manufactured using a 3-D printing technique. Finally, the prototypes were
measured in a planar acquisition range to evaluate the performances of the Bessel beam. Both lenses show
a good agreement between simulations and measurements, obtaining promising results in the Bessel beam
generation by index-graded dielectric lenses at Ka-band.
INDEX TERMS Dielectric lenses, Bessel beams, near-field focusing.
I. INTRODUCTION
Near-field applications have increased their popularity
throughout last years. Most of these applications require
antennas that concentrate the radiated power on a certain area,
typically within their Fresnel region, the radiated near-field
zone. Traditionally, near-field focusing antennas have been
particularly useful in applications such as RFID [1], [2], wireless power transfer (WPT) [3] or medical applications [4].
However, other applications such as near-field radars for
industrial inspection [5] or detection [6], imaging [7], communications [8] or material characterization [9] demand
antennas whose radiated power is not only focused on a
point but on a finite area, increasing the tolerance in the
The associate editor coordinating the review of this manuscript and
approving it for publication was Necmi Biyikli.
VOLUME 8, 2020
probe positioning. A desirable solution for these applications is a near-field focusing beam without diffraction
through its near-field region, thus its transversal profile on the
propagation direction remains unaltered within a significant
range.
Non-diffraction beam solutions have been widely used in
many applications at high frequency bands, particularly at
optical bands, where different techniques have been developed [10]–[12]. Recently, several works have studied new
techniques to generate non-diffraction beams at lower bands,
making the best efforts at millimeter-wave and microwave
bands. A common approach is the design of antennas whose
fields at the aperture are similar to a Bessel function. These
solutions are typically achieved using radial slot arrays
(RLSA) [13]–[15], near-field plates [16] or leaky radial
waveguides [17], [18], among others. An alternative is the
This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/
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use of structures that transform an incident wave to a nondiffraction beam, such as axicons [19], [20], holograms [21]
or meta-surfaces [22]. A new approach is the use of a phase
shifting surface (PSS) [23] or transmitarrays [9] to generate
near-fields with a large depth-of-field at Ka and V bands,
respectively.
Although these previous works obtain good performances
creating non-diffraction beams, they have some drawbacks.
The generation of Bessel beam functions at the antenna
aperture typically require a complex design or manufacturing process. On the other hand, structures like axicons or
holograms involve high costs, complex manufacturing process and bulky solutions. However, planar antenna apertures
(transmitarrays, lenses, PSS . . . ) minimize these issues, being
a suitable alternative in the generation of non-diffraction
beams at millimeter band. In this line, graded-index dielectric
lenses could be considered as a potential candidate to create
these beams too. The working principle of these antennas is
similar to a PSS or transmitarrays but the introduced phaseshift of the incoming wave is controlled by the variation of
the index dielectric media of the cell. One main advantage
regards on the easiness and low-cost fabrication process due
to the use of 3-D printing technology [24]–[26]. This process
is typically applied to planar lenses, where the index dielectric
variation is obtained by adjusting the height of the cells to
obtain the desired phase-shift [27]. However, this technique
does not provide a planar profile in both surfaces therefore,
diffraction problems may appear owing to the edge of the
cells on the non-planar surface.
In this paper, a graded-index only dielectric lens is proposed to generate a near-field with a large depth-of-field, particularly a Bessel beam, at Ka-band. Two novel cells, based
on square and hexagonal prims respectively, are proposed to
physically implement the lens. Both cells are characterized,
and two designs are carried out to evaluate their overall
performances. The designed lenses are manufactured using a
3-D printing technique and measured on a planar acquisition
range. The measured near field shows a good agreement with
simulations.
II. FUNDAMENTALS OF BESSEL BEAMS
IN FINITE APERTURES
The main characteristic of Bessel beams is the constant intensity of the field along the propagation distance [28]. Ideal
Bessel beams are generated by infinite apertures or apertures
whose size is thousands of λ. The amplitude of an ideal Bessel
beam is given by
E (ρ, z) = E0 e−jkz z J0 (kρ ρ)
(1)
where J0 is the zero-order Bessel function, kρ and kz are the
radial and longitudinal components of the free space wave
vector, such that k02 = kρ2 + kz2 .
However, regarding finite apertures, pseudo Bessel beams
are produced. Unlike ideal Bessel beams, the amplitude of
the pseudo Bessel beams only remains constant on a certain
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FIGURE 1. Sketch of the Bessel beam generation through finite aperture
antennas using near-field interferences.
propagation distance before spreading rapidly. The beam is
produced by the generation of a near-field interference pattern
that basically keeps constant within an area. The size of this
area is given by the size of the antenna aperture D.
When this concept is applied at millimeter frequencies
and, particularly to planar lenses, the working principle is
based on the transformation of a spherical incoming wavefront to a plane wave with wave vector lying on a cone,
as Fig. 1 depicts. The outgoing wave front must generate a
near-field interference pattern that behaves as a pseudo Bessel
beam. As Fig. 1 shows, the extent of the shadowed area is
defined by D and γ , and it is called depth-of-field (DoF).
The DoF can be computed using (2) and it also defines the
theoretical 3 dB contour of the field.
D/2
DoF =
(2)
tan (γ )
Theoretically, the maximum distance achievable is limited
by the Fresnel region of the antenna (2D2 /λ). Regarding
the angle γ , it must satisfy sin (γ ) ≪ 1, otherwise the
scalar Bessel beam theory, previously explained, cannot be
applied [29].
Let us consider a planar lens comprised by a given number
of elements regularly distributed on a Nx × Ny grid. The lens
feed is located at a focal length F to generate a Bessel beam
with angle γ . According to Fig. 1, the outgoing wave front
must have its wave vector laying on a cone as long as the wave
front is radiated through the propagation direction ẑ. This
condition allows to create the required near-field interference
pattern to generate the desired beam.
Then, the phase-shift of the lens elements can be computed
using geometrical optics. The phase produced by the elements
of the lens are given by
2π
ϕlens (ρ) = −ϕinc (ρ) − ϕwf −
ρ tan γ
(3)
λ
where ϕwf is the phase of the outgoing wavepfront; ρ is the
axial position of an element computed as ρ = xm2 + y2n , considering the center of the lens the center of the circumference;
and ϕinc (ρ) is the incident phase at an element defined by
q
2π
F 2 + ρ2
(4)
ϕinc (ρ) = −
λ
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
FIGURE 3. Sketch of dielectric cells based on airgaps insertions to control
the effective dielectric constant of the total cell. (a) Square prism airgaps
(b) Cylindrical airgaps.
FIGURE 2. Comparison of Bessel beams using different aperture sizes
assuming ideal lens elements computed with (2) and F = 150 mm and
γ = 5◦ with a cosq feed (q = 7).
Fig. 2 shows a comparison of Bessel lenses axial intensities
for different aperture sizes while keeping γ and F constant.
As expected, the use of larger apertures generates beams more
similar to ideal Bessel beams with a constant intensity on the
propagation direction.
III. DIELECTRIC LENS ELEMENTS
A. WORKING PRINCIPLE OF DIELECTRIC CELLS
A planar lens is required to have the same height in every
cell. One approach that satisfy this condition is using total
dielectric cells. These cells behave as an effective index media
that adds a certain delay to the transmitted ray. The effective
index media ηeff is related to the effective dielectric constant
as n2eff (ρ) = ǫeff (ρ). Thus, the variation of ǫeff allows to
physically implement the phase-shift φlens (ρ) of the cell.
Let us assume two spatially uniform and isotropic materials
with dielectric constants ǫ1 and ǫ2 , respectively. If the second material is embedded in the other, the effective dielectric constant of the assembled cell does not depend on the
geometry but on the dielectric constants and its volume fraction [30]–[32]. The effective dielectric constant of the cell can
be computed as
ǫeff = ǫ1
2ǫ1 + ǫ2 + 2P(ǫ2 − ǫ1 )
2ǫ1 + ǫ2 − P(ǫ2 − ǫ1 )
(5)
where P is the volume fraction of the material ǫ2 over the
total volume of the cell. If one of the dielectrics is air, only
one dielectric is needed to accomplish (5), being ǫ2 = ǫ0 .
In this work, the proposed cells are based on polylactic acid
(PLA) (ǫ1 = 2.85 and tan δ = 0.0121@40 GHz) [33] used
as the host material and air (ǫ2 = 1) to perform the inclusions. Two different cells are analyzed, particularly, square
and hexagonal prism cells. In both cases, a variable airgap
is embedded in the cell to change the volume fraction and
control the φlens (r) that is introduced.
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B. PHASE RESPONSE WITH AIRGAPS
The square prism cells, Fig. 3(a), are based on a square PLA
prism of dimensions a × a × t, and a second embedded airgap
prism of variable dimensions W × W × L as Fig. 3 shows.
The variation of these dimensions changes the volume of air,
therefore P:
P=
Vairgap
W2 × L
= 2
Vprism
a ×t
(6)
The cell is analyzed with CST Microwave Studio [34]
using periodic boundary conditions in x- and y-axis and open
in z-direction. The cell is illuminated by a normal plane
wave propagating in the z-direction with the electric field
defines in the y-direction. The dimensions of the PLA prism
are λd /2 × λd /2 × 2λd (as × as × t), whilst the embedded
prism dimensions are swept between W ∈ (0, λd /2) and
L ∈ [0, 2λd ]. So, either full dielectric cells (ǫeff = 2.85) or
almost air cells (ǫeff = 1.28) are considered. The analysis
is performed at central frequency of 28 GHz in a bandwidth of 2 GHz, and λd is computed as λ0 at the highest
frequency, 30 GHz.
The amplitude and phase response of the transmission
coefficient of the cell as function of the airgap dimensions
are shown in Fig. 4, at the central frequency and normal
incidence. The different combinations of L and W totally
cover the 360◦ required range to implement ϕlens (ρ). More
than 85% of the cells present transmission losses lower than
1.5 dB. These values are expected since the tan δ of PLA
is significantly high. Additionally, Fig. 5 shows the relation
between the dimensions of the airgap and the effective dielectric constant of each cell.
On the other hand, the hexagonal cells, Fig. 3 (b), use
an embedded cylinder airgap. In accordance with (4), this
change in the geometries does not affect the phase-shift so
long as P is the same as the square prism cells. Consequently, the volume of the airgaps, both cylinder and square
prism, must be the same and the volume of the square and
hexagonal prism are also equal. Considering that the lens
depth is independent from the type of selected cell, equals to
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FIGURE 5. Effective dielectric constant of the square prism cell regarding
the volume fraction of air and PLA.
FIGURE 4. Square prism cell response in function of the airgap
dimensions for normal incident at 28 GHz. (a) Phase (deg)
(b) Amplitude (dB).
t = 2λd , the condition is only imposed on the cell prism base.
Therefore, a hexagon of radius ah = 0.31λd is obtained.
In order to compute the cylinder dimensions, the depth
of the cylinders remain alike to the depth of square prism
airgaps, L. Thus, only the radius needs to be calculated using
R2 = W 2 /(π ). In Fig. 6, the amplitude and phase of the transmission coefficient versus frequency for different square and
hexagonal prisms are compared, showing that the geometry
does not affect the cell response.
C. ANGULAR PERFORMANCES
The phase stability of the cell has been analyzed for different
angles of incidence θ and φ at 28 GHz. It was considered
linear polarization of the incident electric field and angles
with variation in φ = 0◦ , 45◦ and 90◦ and θ ∈ [0, 30]◦ ,
being θ = 0◦ normal incidence. Fig. 7 shows the phase
responses from a set of 6 different cells as a function of
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FIGURE 6. Comparison of the (a) transmission phase and
(b) transmission amplitude of the different prisms with the same filling
factor. Airgap dimensions are in mm.
θ and φ. These results show the independence to φ, whilst θ
notably modifies the phase-shift regarding normal incidence.
The dependence of phase-shift versus θ can be approximated
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
FIGURE 8. Cell distribution of the lens elements (left) Hexagon prism
cells (right) Square prism cell.
FIGURE 7. Phase response analyzed for different angles of incidence and
the approximation obtained by a polynomial expression for different cells
at 28 GHz. (solid) φ = 0◦ (Grey square) φ = 45◦ (Green circle) φ = 90◦ .
Airgap dimensions are in mm.
by a second-order polynomial as:
1ϕ (θ ) = p1 · θ + p2 · θ 2
(7)
As shown in Fig. 7, the curve shape is very similar to the
analyzed cells and the polynomic model of 1ϕ in the previous
equation is common for all of them (p1 = 0.7453 and
p2 = 0.0425). Then, the phase of the transmission coefficient
for each cell of the lens can, therefore, be determined using
the following equation.
ϕlens (θ ) = ϕnorm + 1ϕ(θ)
The proposed antenna should generate the Bessel beam
close to the lens, therefore a maximum DoF of 650 mm is
established. According to (2), the beam theoretically behaves
as a Bessel beam within this area if γ = 5◦ .
B. DESIGN PROCEDURE
The lens design is based on geometrical optics theory according to (3) and (4). These equations provide the required phaseshift that should introduce the lens elements to radiate the
beam previously defined. In this case, the phase distribution
of the elements along the lens surface is shown in Fig. 9,
reminding the physical distribution shown in Fig. 8.
(8)
being 1ϕ obtained with the polynomial (6) and ϕnorm the
phase of the transmission coefficient obtained for normal
incidence analysis. This function nearly predicts the real
phase-shift of a cell under oblique incidence, without considering the angle on the preliminary cell study. The maximum
error produced in most cases is lower than 5◦ .
IV. DIELECTRIC BESSEL LENS BASED ON AIRGAP CELLS
In order to validate both square and hexagonal prism cells,
this section addresses the lens design and simulation for both
type of cells.
A. ANTENNA OPTICS
The proposed antenna to generate a Bessel beam is a
square planar dielectric lens of equivalent aperture D of
120 × 120 mm2 at 28 GHz. The distribution of the elements
depends on the cell geometry. For square cells, a regular
grid is used with periodicity 5 × 5 mm2 and made up of
24 × 24 elements. In the case of the hexagonal prism cells,
they are arranged as a honeycomb, so the cells are located on
an axial distribution as shown in Fig. 8.
The lenses are defined using a centered optics (see Fig. 1),
and the phase center of the feed is placed at (xf , yf , zf ) =
(0, 0, F = −100) mm, taking the center of the lens as the
origin of the system of coordinates.
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FIGURE 9. Transmission phase-shift (deg) of the lens element along the
surface to generate the desired Bessel beam.
Once the phase distribution along the lens surface is computed, the elements must be designed. In the designing process, the dimensions of the airgaps (R, W and L) are adjusted
to produce the required phase-shift according to φlens (ρ) in
equation (3). Note that for, each cell, the incidence angle (θ),
the focal distance (F) of the lens and the radius of the center
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
of the cell in the lens (ρ) are related through
ρ
θ = atan
(9)
F
Therefore, the equation (6) can be rewritten in the following form:
ρ
ρ 2
1ϕ (ρ) = p1 · atan
+ p2 atan
(10)
F
F
And then substituting (10) in (8)
ϕlens (ρ) = ϕnorm + 1ϕ (ρ) = ϕnorm
ρ 2
ρ
+ p2 atan
(11)
+ p1 · atan
F
F
Finally, the adjustment is done element by element, considering the real angle of incidence ϕnorm shown in Fig. 4(b)
with the desired one ϕlens (ρ) of Fig. 9. In order to improve the
design, an angular correction 1ϕ(ρ) is introduced throughout
the process; thus, the real phase-shift of each cell is considered. The resulting lens layout is made by the cells that
minimize their phase-shift to the theoretical response. The
amplitude of the cell response is used to discard cells with
high losses.
FIGURE 11. Depicted setup of the full-wave simulation carried out in CST
Microwave Studio. A half quarter of the lens has been removed to
observe the airgap inclusions.
FIGURE 12. XoZ plane of the Bessel beam created by the lens using
(a) square prism cells (b) hexagonal prism cells.
FIGURE 10. Cut y = 0 mm of the phase-shift (deg) of the lens element
along the lens surface to generate the desired Bessel beam.
In Fig. 10 the cut y = 0 mm of the theoretical phase
distribution is compared with the chosen cell of the final
design, showing a good agreement between both. Additionally, a comparison with the design, based only on normal
incidence, is done to highlight the importance of the angular
correction through the design process. Notice that the difference between selected cells occurs more at the edge of the
lens where the incidence angle is farther from the normal
incidence. Two different layouts are obtained, one for each
type of cell.
In the evaluation of the Bessel beam performances, both
layouts have been analysed with CST Microwave Studio [32]
in a full-wave simulation (see Fig. 11). The selected feed
is a pyramidal horn antenna with 15 dBi gain placed at
a distance F = 100 mm from the surface of the lens.
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This configuration illuminates the lens surface with an amplitude taper of −17 dB at the edge of the lens.
The simulation results are shown in Fig. 12 for the XoZ
plane and in Fig. 14 the electric field along the z-axis is
compared with the measurements. Because the horn antenna
is non-symmetric regarding its main cuts, E- and H - plane,
the outgoing wave-front is not perfectly formed. Hence, the
near-field interference pattern is not properly created in the
closest area of the antenna, having strong fluctuations on this
area. This effect is minimized when hexagonal prism cells
are used in the design. The cells achieve a more stable beam
and concentrate it through a larger range than square prism
cells, as Fig. 12 shows. However, when a horn antenna feeds
a Bessel beam generator with small aperture, it is expected to
obtain a shorter DoF range than the one computed with (2).
V. EXPERIMENTAL RESULTS
Both planar dielectric lenses, made up of either square or
hexagonal cells, have been manufactured and evaluated in the
planar acquisition range at University of Oviedo. The lenses
have been manufactured using a Fused Deposition Modeling (FDM), a 3-D printing technique based on the melting
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
FIGURE 13. (a) Manufactured prototypes using 3-D printing techniques
(FDM). Lens made up of hexagon prism cells (left) and lens made up of
square prism cells (right). (b) Setup at the planar acquisition range of
University of Oviedo to measure the planar dielectric lens of hexagon
prism cells at Ka-band.
FIGURE 14. Comparison between simulations and measurements of the
normalized amplitude (dB) of the electric field along z-direction and
x = y = 0 at 28 GHz. (a) Square prism cell lens (b) hexagonal prism cell
lens.
and extrusion of a thermoplastic polymer, such PLA, through
a nozzle tip to deposit the material layer-by-layer onto a platform. This technique is widely used in additive manufacturing
processes to fabricate prototypes since its easily controlled
and reduces the costs significantly. However, when dealing
with accurate designs likes these lenses, it is important to set
a proper configuration to obtain a high precision. Especially,
the thickness wall and layer height since both control the
resolution of the pieces. In this case, the thickness wall is set
to 0.4 mm and the layer height to 0.1 mm, providing a highresolution printing. Low resolution configuration could modify or eliminate the internal wall of low infill cell (P factor
close to 1), therefore change the effective dielectric constant
and the cell response. On the other hand, high-resolution configurations inherently increase the printing time to 40 hours
for the lens made of square prism cells and 48 hours for the
lens made of hexagon prisms cells. Throughout the whole
process is highly relevant to keep the environmental conditions, for instance variation on temperature could produce air
flows and lead to structural deformations on the pieces such
as curvatures on the corners.
Both prototypes are shown in Fig. 13(a) and Fig. 13(b)
depicts the setup used to measure both lenses. In this setup
a vector network analyzer (PNA-X of Keysight) is connected to the feeding horn, a pyramidal standard gain horn
of 15 dBi gain, whilst the second port is connected to the
probe, an open-ended Ka-band waveguide. The lens is placed
on a PLA structure aligned to the aperture of the probe. The
measurements are performed from 26 to 30 GHz evaluating
the electric field at the horizontal plane XoZ and at different
transversal planes XY .
The normalized electric field along the propagation direction ẑ is compared with simulations in Fig. 14 at 28 GHz.
Measurements highly agreed with simulations, highlighting
the range of the DoF or the decay of the Bessel beam.
Additionally, the transversal XoZ (y = 0) plane is shown
in Fig. 15 at different frequencies for both lenses. The
3 dB level indicates the maximum attenuation allowed in
the DoF, whose theoretical range is 650 mm according
to (2). Both lenses obtain their best results at the highest
frequencies, from 28 to 30 GHz, where the DoF reaches
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
FIGURE 15. Normalized measured amplitude (dB) of the (a) square cell (b) hexagon cell lens at different frequencies at the plane xoz (y = 0).
a 475 (47.50λ0 ) mm range. It is worth noting the hexagonal
prism lens behavior, showing an improvement in the results
regarding the square prism cell. The in-band response, particularly at the upper frequencies, barely change, whilst square
prism lens increases its variation due to frequency shifts.
Furthermore, the non-diffraction area is larger in all cases, but
it also starts closer to the antenna aperture, creating a more
stable Bessel beam through the propagation direction.
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The transverse profile is also evaluated in a plane XY at
z = 250 mm at 26.5, 28 and 29.5 GHz. The plane is parallel
to the antenna aperture and shown in Fig. 16. Measurements
show good axial symmetry with a field distribution similar to
the Bessel main lobe, whilst the side lobes are hidden due to
the antenna size.
The cut y = 0 of three different transversal XY planes,
z = 150, 200 and 250 mm, is measured at the whole band and
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
FIGURE 16. Measured transversal XY profiles at z = 250 mm parallel to the antenna aperture at 26.5, 28 and 29.5 GHz.
TABLE 1. Comparison of the antenna and Bessel beam performances with published works.
shown in Fig. 17. These measurements evaluate the beamwaist of the beams through the non-diffraction range. Neither
the hexagonal prism lens nor the square prism lens is able
to properly form the beam at the closest plane, z = 150 mm.
However, at z = 200 mm the beam of the hexagonal
prism lens is similar to a Bessel distribution and the beam
is confined in a spot smaller than 4λ. The main lobe of
the square prism lens is flattened at this plane, especially
at 28 GHz and higher frequencies. At the further plane,
z = 250 mm, the square prism lens shows a beam similar
to a Bessel function. As expected, the hexagonal prism lens
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keeps its Bessel behaviour. It must be noted that the hexagonal beam-waist nearly unchanges when the frequency varies.
Different published works of 3-D printed Bessel lens are
compared in Table 1. Regardless the geometry and frequency,
these works use cells based on a height variation to control
the phase-shift introduced by each cell. In this work, the cell
keeps its height constant and the phase-shift is obtained by
the insertions of airgaps, obtaining a double planar surface
lens that minimizes possible diffractions caused by the height
difference of the cells and offers an easier integration of the
lens. Considering the size, the depth-of-field and, especially
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FIGURE 17. Normalized amplitude (dB) at (left) z = 150 mm (center) z = 200 mm and (right) z = 250 mm and y = 0 and different
frequencies.
working in Ka-band, the results of this work make the proposed cells as an attractive solution in the generation of Bessel
beams using 3-D printing technology in millimeter band.
VI. CONCLUSION
Dielectric planar lenses are demonstrated to generate nearfield Bessel beams in Ka-band. The proposed lenses are made
up of dielectric cells based on either square or hexagonal
prisms. Both cells ensure the variation of their index dielectric
media using airgaps to control the overall density of the
material. A design is carried out to generate a Bessel beam
with a depth-of-field of 650 mm, and it has been implemented
for both type of cells, each one based on a proposed cell. The
dependence of the phase response of the cell and the angle of
incidence is considered with a second order polynomial. Both
prototypes were manufactured using a 3-D printing technique
and measured in a planar acquisition range, obtaining a good
agreement between simulations and measurements. Although
measurements show good performances in both cases, the
hexagonal prism cell exhibits a superior behavior regarding
the non-diffraction range and the in-band response. In light
of these results, graded-index dielectric lenses have demonstrated to be a potential candidate to generate Bessel beams
at Ka-band frequencies, taking the advantage of reducing the
manufacturing cost process and reaching a simple structure.
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MARCOS RODRIGUEZ PINO was born in Vigo,
Spain, in 1972. He received the M.Sc. and Ph.D.
degrees in telecommunication engineering from
the University of Vigo, Vigo, in 1997 and 2000,
respectively.
In 1998, he was a Visiting Scholar with the ElectroScience Laboratory, The Ohio State University,
Columbus, OH, USA. From 2000 to 2001, he was
an Assistant Professor with the University of Vigo.
Since 2001, he has been with the Electrical Engineering Department, University of Oviedo, Gijón, Spain, where he is currently an Associate Professor, teaching courses on communication systems
and antenna design. His current research interests include antenna design,
measurement techniques, and efficient computational techniques applied to
EM problems, such as evaluation of radar cross section or scattering from
rough surfaces.
ÁLVARO F. VAQUERO (Student Member, IEEE)
was born in Salinas, Spain, in 1990. He received
the B.Sc. and M.Sc. degrees in telecommunications engineering from the Universidad de Oviedo,
Gijón, Spain, in 2015 and 2017, respectively,
where he is currently pursuing the Ph.D. degree.
Since 2016, he has been a Research Assistant
with the Group of Signal Theory and Communications, University of Oviedo. In 2017, he was
with the Instituto de Telecomunicações, Lisbon,
Portugal, where he was involved in broadband planar lenses for skin cancer
imaging. His current research interests include the development of efficient
techniques for the analysis and synthesis of reflectarrays and planar lenses
and the design of 3-D printed lenses for near-field applications.
SÉRGIO A. MATOS (Senior Member, IEEE)
received the Licenciado, M.Sc., and Ph.D. degrees
in electrical and computer engineering from the
Instituto Superior Técnico (IST), University of
Lisbon, Lisbon, Portugal, in 2004, 2005, and 2010,
respectively.
He is currently a Researcher with the Instituto de Telecomunicações (IT), Lisbon. He is also
an Assistant Professor with the Departamento de
Ciências e Tecnologias da Informação, Instituto
Universitário de Lisboa (ISCTE-IUL). He has coauthored 80 technical papers
in international journals and conference proceedings. His current research
interests include electromagnetic wave propagation in metamaterials, flatlens design, and transmit arrays.
VOLUME 8, 2020
MANUEL ARREBOLA (Senior Member, IEEE)
was born in Lucena (Córdoba), Spain. He received
the M.Sc. degree in telecommunication engineering from the University of Málaga, Málaga, Spain,
in 2002, and the Ph.D. degree from the Technical University of Madrid (UPM), Madrid, Spain,
in 2008.
From 2003 to 2007, he was a Research Assistant
with the Electromagnetism and Circuit Theory
Department, UPM. In 2005, he was a Visiting
Scholar with the Microwave Techniques Department, Universität Ulm, Ulm,
Germany. In 2007, he joined the Electrical Engineering Department, University of Oviedo, Gijón, Spain, where he is currently an Associate Professor.
In 2009, he enjoyed a two-month stay at the European Space Research
and Technology Centre, European Space Agency, Noordwijk, The Netherlands. In 2018, he was a Visiting Professor with the Edward S. Rogers Sr.
Department of Electrical and Computer Engineering, University of Toronto,
Toronto, ON, Canada. In 2019, he was a Visiting Professor with the Institute
of Sensors, Signals and Systems, Heriot-Watt University, Edinburgh, U.K.
His current research interests include the development of efficient analysis,
design, and optimization techniques of reflectarray and transtmitarray antennas both in near and far fields.
Dr. Arrebola was a co-recipient of the 2007 S. A. Schelkunoff Transactions
Prize Paper Award by the IEEE Antennas and Propagation Society.
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Á. F. Vaquero et al.: Bessel Beam Generation Using Dielectric Planar Lenses at Millimeter Frequencies
JORGE R. COSTA (Senior Member, IEEE) was
born in Lisbon, Portugal, in 1974. He received
the Licenciado and Ph.D. degrees in electrical and
computer engineering from the Instituto Superior
Técnico (IST), Technical University of Lisbon,
Lisbon, in 1997 and 2002, respectively.
He is currently a Senior Researcher with the
Instituto de Telecomunicações, Lisbon. He is also
a Full Professor with the Departamento de Ciências e Tecnologias da Informação, Instituto Universitário de Lisboa (ISCTE-IUL). His current research interests include
lenses, transmit-arrays, and biomedical antennas. He is the coauthor of
four patent applications and more than 200 contributions to peer-reviewed
journals and international conference proceedings. More than 40 articles
have appeared in IEEE journals.
Dr. Costa was the Co-Chair of the Technical Program Committee of
the European Conference on Antennas and Propagation (EuCAP 2015),
Lisbon, and the General Vice-Chair of EuCAP 2017, Paris. He served as an
Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION,
from 2010 and 2016. He is currently an Associate Editor of the IEEE OPEN
JOURNAL OF ANTENNAS AND PROPAGATION. He was a Guest Editor of the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION Special Issue on Antennas and
Propagation at mm- and submm-Waves in 2013.
216196
CARLOS A. FERNANDES (Senior Member,
IEEE) received the Licenciado, M.Sc., and Ph.D.
degrees in electrical and computer engineering
from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal,
in 1980, 1985, and 1990, respectively.
In 1980, he joined IST, where he is currently
a Full Professor with the Department of Electrical and Computer Engineering in the areas of
microwaves, radio wave propagation, and antennas. He is also a Senior Researcher with the Instituto de Telecomunicações
and a member of the Board of Directors. He has coauthored a book, two book
chapters, and more than 180 technical papers in peer-reviewed international
journals and conference proceedings. He holds seven patents in the areas of
antennas and radiowave propagation modeling. His current research interests
include dielectric antennas for millimeter-wave applications, antennas and
propagation modeling for personal communications systems, RFID and
UWB antennas, artificial dielectrics, and metamaterials.
Dr. Fernandes was a Guest Editor of the IEEE TRANSACTIONS ON ANTENNAS
AND PROPAGATION Special Issue on Antennas and Propagation at mm- and
submm-Waves in 2013.
VOLUME 8, 2020