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Abstract—Future sub-millimetre imagers are being developed
with large focal plane arrays (FPAs) of lenses to increase the field
of view (FoV) and the imaging speed. A full-wave electromagnetic
analysis of such arrays is numerically cumbersome and time-
consuming. This paper presents a spectral technique based on
Fourier Optics combined with Geometrical Optics for analysing,
in reception, lens based FPAs with wide FoVs. The technique
provides a numerically efficient methodology to derive the Plane
Wave Spectrum (PWS) of a secondary Quasi Optical component.
This PWS is used to calculate the power received by an antenna or
absorber placed at the focal region of a lens. The method is applied
to maximize the scanning performance of imagers with
monolithically integrated lens feeds without employing an
optimization algorithm. The derived PWS can be directly used to
define the lens and feed properties. The synthesized FPA achieved
scan losses much lower than the ones predicted by standard
formulas for horn based FPAs. In particular, a FPA with scan loss
below while scanning up to (beam-widths) is
presented with directivity of complying with the needs for
future sub-millimetre imagers. The technique is validated via a
Physical Optics code with excellent agreement.
Index Terms— Focal Plane Arrays (FPAs), Reflector Antennas,
Lens Antennas, Fourier Optics, Geometrical Optics, Spectral
Techniques, Sub-millimetre Wavelengths.
I. INTRODUCTION
EW generations of imaging cameras at (sub)-millimetre
wavelengths are emerging [1]−[8]. Large format fly’s eye
lens arrays coupled to antennas or absorbers based detectors are
being developed for these cameras. For instance, cryogenic
Kinetic Inductance Detectors (KIDs) coupled lenses are
employed for passive cameras [3], [6]. Moreover, future high
frequency communication systems will use integrated lens
antenna technology [9]. The next generation of sub-millimetre
imagers are planned to have focal plane arrays (FPAs) with over
1000 detectors to improve the overall image acquisition speed.
In all these scenarios, a full-wave electromagnetic analysis,
which includes the coupling between the quasi-optical (QO)
system and the detector array, is numerically cumbersome and
time-consuming. A typical approach for analysing such
coupling in transmission resorts to the use of the Physical
Optics (PO) and simplified Geometrical Optics (GO) based
This work was supported by the European Research Council Starting Grant
(ERC-2014-StG LAA-THz-CC), No. 639749.
S. O. Dabironezare, G. Carluccio, A. Neto and N. Llombart are with the THz
Sensing Group, Microelectronics Department, Delft University of Technology,
2628 CD Delft, The Netherlands (e-mail: dabironezare.shahab@ieee.org).
techniques for antennas [10] and absorbers [11], respectively.
In this paper, we propose, the characterization of wide field
of view imagers via the derivation of their plane wave spectrum
(PWS) in reception. The approach simplifies the design of the
lens focal plane array since both the lens shape and the feed
radiation properties can be derived directly from the PWS
without the need of using an optimization algorithm. The
optimal radiation pattern of an antenna feed can then be directly
derived by applying a conjugate field match condition [12]. In
the case of absorbers, their optimal angular response can be
derived by linking the PWS to an equivalent Floquet mode
circuit as in [13].
In [14], a numerical evaluation of the incident PWS in a
reflector system was described. A much simpler approach using
Fourier Optics (FO) was proposed in [13] and [15]. Over a
limited applicability domain, the later approach leads to
analytical expressions for the PWS for specific geometries for
broadside or slightly squinted incident angles. In this work, we
extend the FO approach for quasi-optical systems with multiple
components and wide-angle applications by combining it with
a numerical GO based technique in reception. The analyses in
[13] and [15] were aiming to focal plane arrays of bare
absorbers. Therefore, the derived PWS has not taken into
account the quadratic dependence of the focal field phase. Here,
to properly include the coupling between two QO components
in the PWS field representation, especially for off-focuses
cases, the quadratic phase is efficiently introduced by applying
a local phase linearization around the observation point in the
focal plane.
The developed technique is then applied to the synthesis of a
wide field of view imager complying with the needs for future
sub-millimetre imagers for security applications [8], [16]−[18].
For these applications antenna gains of about of to dBi are
required [8] with about beams.
Various solutions have been proposed in the past to improve
the scanning performance of quasi-optical systems either using
Gaussian horn feeds combined with shaped reflector or lens
antennas [19]−[22] (with most of the cases over sizing the
radiating aperture) and/or determining an optimum focal
surface [23], [24], where the array elements are placed [17]; or
A. Freni is with the with the Dipartimento di Ingegneria dell’Informazione,
University of Florence, 50139 Florence, Italy (e-mail: angelo.freni@unifi.it).
Coherent Fourier Optics Model for the
Synthesis of Large Format Lens Based Focal
Plane Arrays
Shahab Oddin Dabironezare, Student Member, IEEE, Giorgio Carluccio, Member, IEEE, Angelo
Freni, Senior Member, IEEE, Andrea Neto, Fellow, IEEE, and Nuria Llombart, Fellow, IEEE
N
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by using array clusters of feeds to achieve a conjugate field
match condition with the focal plane field [25]−[27]. This work
considers a relatively simple FPA architecture based on a lens
array. All the lens feeds are placed over a flat surface, enabling
monolithic integration at high frequencies. The surface shape of
the lenses is linked directly to the phase of the incident PWS,
while the radiation of the lens feeds is matched to the amplitude
of the PWS via a Gaussian model approximation. For
simplicity, the main reflector aperture is modelled as a
symmetric non-oversized parabola. The obtained
performances, validated via a conventional PO analysis, show
significantly lower scan loss than it would be obtained by
placing Gaussian horns in the optimal focal surface of such
reflector as in [17]. The proposed technique could be easily
extended to more practical reflector implementation (e.g. a
Dragonian dual reflector [28]) by linking the PWS derivation to
a GO field propagation in the reflector system and adjust
accordingly the lens surfaces, as well as in combination with
oversized shaped surfaces.
The paper is structured as follows. Section II describes the
proposed FO/GO methodology to derive a PWS field
representation in a multi-cascade quasi-optical system, while
Section III extends this technique to wide-angle optics. In
Section IV, the methodology is applied to a Fly’s eye lens array,
and Section V presents an application case. Concluding remarks
are given in Section VI.
Fig. 1. Coherent FO scenario: a focusing QO component is illuminated by an
incident plane wave. A PWS representation of the focal field impinging on a
secondary QO component (shown in the inset) located at . The local
reference system at the neighbourhood of is also shown.
II. COHERENT FOURIER OPTICS
In this section, a plane wave spectrum (PWS) representation,
for the magnitude and phase of the focal field is developed. This
PWS is derived for a generic QO component illuminated by a
plane wave, using a new coherent FO approach. In [15], the
PWS represented only the magnitude of the focal field, since
the effort was focused on analysing incoherent detectors.
Conversely, including the phase in the PWS is now essential for
accurately representing the coupling between multiple QO
components, depicted in the scenario shown in Fig. 1, or for
evaluating the performance of a QO system with a coherent
detection scheme. The phase can be efficiently introduced in the
PWS by applying a local linearization as shown in this section.
Let us consider a generic focusing QO component
illuminated by a plane wave
, with wave-
vector
. As shown in Fig. 1, an equivalent FO sphere centered
at the focus of the component can be used to represent the direct
field, , on the focal plane () in terms of a PWS
([15], [13]):
(1)
where is the radius of the equivalent FO sphere,
, with being the wave-number of the medium
surrounding the focal plane, and
is the PWS of the
direct field. The last quantity can be calculated as follows [13]:
, (2)
where
, and
is the
GO field component tangent to the equivalent FO sphere. This
GO field is defined over the angular sector subtended by the
optical system ( in Fig. 1). This GO field can be calculated
analytically [13] when a parabolic reflector or elliptical lens is
illuminated by a slightly skewed incident plane wave (
). For larger illumination angles and for a generic QO
component, a numerical GO based approach can be employed
[29]. Specifically, the field over its FO sphere can be expressed
as follows:
(3)
where, represent a scenario involving a transmitting (e.g.
lens) or a reflective (e.g. mirror) surface, respectively;
is the incident field evaluated at the point of the QO surface
(see Fig. A.1 in the Appendix); and
are the transmission and reflection dyads,
respectively; () and () are the perpendicular and
parallel transmission (reflection) coefficients on the surface,
respectively; (
) represents the polarization unit vector
of the transmitted (reflected) rays;
and
are the
principal radii of curvature of the transmitted/reflected wave
fronts; is the length of the GO ray propagating from the QO
component to the FO sphere, Fig. 1. The expression of the GO
parameters in (3) for the transmission case is provided in the
Appendix. As for the detailed derivation of the reflection and
refraction cases, the reader is addressed to [30].
The integral in (1) resembles an inverse Fourier transform
which relates the spectral field
to the spatial one, ,
except for the presence of the quadratic phase term,
. As an example for demonstrating the
importance of including the quadratic phase term into the PWS
representation, let us consider a parabolic reflector with a
diameter of , and a f-number . The
reflector is assumed illuminated by a polarized plane
wave with . The same scenario is going to be
analysed throughout this paper. As an example here, an incident
Focusing QO
Component
FO sphere
PWS
FO
applicability
region
PWS
FO
sphere
Incident Plane
Wave
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angle , i.e. scanning the reflector by beams, is
considered. The corresponding variation of the quadratic phase
term is shown in Fig. 2(a). The position of the geometrical
flashpoint, is also shown. We define the geometrical flash
point as the position of the peak of the focalized field over a
focal plane assuming that no higher order aberrations are
present, i.e. the beam deviation factor (BDF) is 1. Figure 2(b)
shows the magnitude of the LHS of (2) along
when
. By considering the quadratic phase term constant and
equal to the one taken at the flashpoint (i.e.
) the
spectrum is flat over the reflector spectral domain (solid black
line). ). Figures 2(c) and (d), show the magnitude and phase
values of the focal field , respectively, along the x-axis in a
region close to the flash point. The result obtained when
assuming constant quadratic phase term (solid black line), as in
[15], are compared against a reference solution using a standard
PO based code (dotted blue line). It is evident that the
magnitude of the focal field is accurately represented, but the
phase is not.
To properly represent the phase, we can rewrite (1) as the
product of two spatial functions:
(4)
where
is the quadratic phase term, and
represents the inverse Fourier transform of
. The spatial field in (4) can then be expressed as an
inverse Fourier transform of the function
, referred
here as the coherent FO (CFO) spectrum:
. (5)
Specifically, the CFO spectrum is given by:
, (6)
where is the convolution operator, and
is the Fourier
transform of the quadratic phase term, which can be expressed
analytically as:
. (7)
With reference to the previous example, the grey curve of
Fig. 2(b) shows the variation of the magnitude of the coherent
FO spectrum. The spectrum is now highly oscillating and
shifted with respect to the one of the FO approximation. In Figs.
2(c) and (d), it is shown that both the magnitude and phase of
the focal field are accurately calculate using (5). However, due
to the oscillations present in the convoluted spectrum (see Fig.
2(b)), the numerical convergence of the integral in (5) is more
demanding than the one in (1).
We can simplify the calculation of the coherent FO spectrum
by approximating the quadratic phase term with a linear one
which accurately represents the field only at the surrounding of
a specific position in the focal plane. This position is referred to
as the CFO position, . This approximation is achieved by
introducing a new reference system in the focal plane centred at
this position, where (Fig. 1), and retaining the
first two terms of the Maclaurin series of the quadratic phase
argument:
. (8)
(a)
(b)
(c)
(d)
Fig. 2. A parabolic reflector with and illuminated by a
plane wave with an incident angle of : (a) quadratic phase
term, (b) FO spectrum. The insets illustrate the 2-D spectrum of the -
component of the field, where left, middle, and right panels represent the
,
convoluted spectrum, and the shifted one, respectively. (c) Magnitude, and (d)
phase of the electrical focal field. The grey region indicates the applicability
region of linearization approximation as stated in (12).
The result of this linearization is shown in Fig. 2(a) (dash red
line), where is chosen. The Fourier transform of the
quadratic phase term,
, can be approximated as
, (9)
where
. Therefore, the convolution operation in
(6) simply results in a shift of the FO spectrum along
.
Quadratic Phase [deg]
0
50
400
150
200
100
-12 12
-4 410
068
2
-2-6
-10 -8
300
250
350
-0.2 2
(dB)
-90
-80
-75
-65
-0.15 -0.5 1-0.1 0
-70
0.5 1.5
-85
111
(dB)
-25
-15
-5
35
2 5 83 6
15
7 9
25
5
104
PO
111
[deg]
-200
-100
0
200
2 5 83 6
100
7 9
150
50
104
-50
-150
PO
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Then, the focal field can be evaluated in the new reference
system via
, (10)
where the coherent FO spectrum is approximated as follows:
(11)
When examining Fig. 2(b), we can notice that the
approximated coherent FO spectrum (dash red line) has a
spectral domain similar to the one calculated by using (6), but
without oscillations. In Figs. 2(c) and (d), both magnitude and
phase of the spatial field are reported (dash red line),
respectively. The agreement of the obtained results with the one
of the PO solution (dash blue line) is evident. The grey region
shown in these figures corresponds to the applicability region
of the approximation (8). This region is defined as a circle,
centred on , with diameter
where a maximum phase
error of is allowed in the quadratic phase term:
, (12)
where and are the diameter and the f-number of the
corresponding QO component, respectively. It is worth noting
that this applicability region does not depend on the chosen
CFO position, . Figure 3(a) shows the diameter of this
applicability region for a few f-number cases of a parabolic
reflector versus its diameter . It can be noted that as the
reflector f-number increases, the number of beams that could be
analysed using (8) decreases. For comparison, the dashed
curves in the figure show the applicability region of the
spectrum in (1) when a constant quadratic phase evaluated at
is considered. In the latter case, applicability region
depends on the chosen , and the approximation can only be
used for a region close to the origin, and small .
Fig. 3. (a) Applicability region of phase linearization approximation for
different parabolic reflector f-numbers versus its linear dimension. The solid
and dashed curves correspond to the phase linearization, and a constant phase
at approximations, respectively. The latter limit is shown for a parabolic
reflector scanning to beams. (b) Validity region of the FO method, when
analysing a parabolic reflector with diameter , versus the position
of the FO sphere centred in the focal plane.
The diffractive coupling between a primary QO component
and a secondary one, as shown in Fig. 1, can be represented
using the PWS in (11). The focal field of this secondary QO
component can also be represented using (1) and (2). In this case,
the GO field at the FO sphere of the secondary QO component,
, is calculated by propagating each incoming plane
wave from the spectrum of the primary QO component to the
FO sphere of the secondary component. As a result,
is the summation of the contribution of each plane wave from
the spectrum of the primary component.
Coupling of the QO System to Antenna Feeds
Once the PWS of a QO system is derived, the coupling to
antenna based feeds can be analysed resorting to a reception
formulation [12] where the equivalent Thévenin open circuit
voltage of each antenna can be evaluated as follows:
. (13)
is the far field radiated to the FO sphere, by the antenna
when equivalently fed by a current of ; and is the wave
impedance of the medium in which the antenna is embedded.
The power delivered to the load of the receiving antenna can
be calculated as , being the
total power radiated by the antenna when fed with the current
. The power delivered to the feed is maximized when its far
field is equal to the conjugate of the CFO spectrum. This
condition is referred to as the conjugate field match condition.
After calculating the power delivered to the load, one can
estimate the aperture efficiency of the entire QO system as
, where ; is the free
space impedance, is the amplitude of the plane wave incident
on the main QO component, and is its physical area.
By using reciprocity, the electric field,
, that the same
antenna feed would radiate in , at a far distance
from the QO system, can be evaluated as follows:
(14)
where
are the induced Thévenin open circuit voltages,
(13), for a TM/TE polarized plane wave impinging on the main
QO component from the .
III. WIDE-ANGLE OPTICS
The method reported in Section II can accurately represent
the PWS of a QO component within the FO applicability region
introduced in [15]. However, this region limits the maximum
size of a FPA under analysis. In this section, we extend the CFO
method derived previously to cases where the FPA is larger
than this applicability region.
For this purpose, we divide a large FPA into sub-regions
where at the centre of each sub-region a FO sphere (off-centred)
is placed, as shown in Fig. 4. The GO field is then evaluated
over the new sphere. The maximum subtended angle of the
sphere (in Fig. 4) is then defined by the region where GO
field exists. Once the centre of the reference system is changed
to
, identical steps to the ones described in Section II can be
taken to derive the PWS.
The validity region of the FO representation is directly
proportional to the radius chosen for the FO sphere [15].
Moreover, the field tangent to an off-focus FO sphere can be
approximated by using the GO ray propagation when the
surface of the sphere is far from the caustic points of the
geometry (where the focal field is maximum). Specifically, the
GO representation is accurate when the radius of the off-focus
0
0
12
20
400 1000800200 600
16
4
8
0150
0
30
25 10050 125
15
25
10
75
20
5
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FO sphere is greater than the depth of focus,
, where is the Fresnel number.
Therefore, to maximize this radius, it is convenient to define
as the z-position where the off-focus sphere is tangent to the
surface of the QO component (Fig. 4). For this case, the
maximum rim angle for the m-th off-focus sphere can be
expressed as follows
, (15)
where
indicates the distance of the centre of off-focus FO
sphere, in the focal plane, from the QO component origin.
Fig. 4. A schematic representation of the m-th off-focus FO sphere placed in
the focal plane of a focusing QO component.
As example cases, we considered a parabolic reflector with
and , or one with , illuminated by
a plane wave impinging with an angle far from the broadside.
The normalized focal field for these two cases is shown in
Fig. 5, and compared, with an excellent agreement, with a
standard PO based approach.
The FO representation of the focal fields can be derived by
performing approximations in the PO radiation integral as
described in [15]. Specifically, approximations on magnitude,
vector, and phase terms in the integrand. The applicability
region for the FO method can then derived by imposing a
maximum acceptable value for the error committed in these
approximations, for the magnitude and vector cases, and
for the phase. By following the same steps as in [15], for the m-
th off-focus FO, one can define the following validity regions:
. (16)
Figure 3(b) shows the validity region of the off-focus FO for
reflectors with different f-numbers assuming and
as in [15]. As it can be easily seen, the broadside FO
validity region is larger for greater f-numbers. However, for
reflectors with large f-number this region decreases more
rapidly as the sphere is farther away from the focus. This is due
to the fact that grows rapidly when the reflector f-
number is large. It is worth noting that following similar steps,
one can derive the FO applicability region in the vertical
direction with respect to the focal plane ( in Fig. 4) as
described in [30]. This vertical applicability region can be
extended further by displacing the center of the equivalent FO
sphere in the -direction. This extension leads to the possibility
of analyzing non-focal plane arrays such as imaging reflector
antennas for satellite communications [31].
(a) (b)
(c) (d)
Fig. 5. Magnitude and phase of the x-component of the electric field at the focal
plane of a parabolic reflector with and (a)-(b) , or (c)-(d)
. The plane wave impinging angle is , and the off-focus
FO sphere is placed at
. Grey region indicates the applicability
region of the FO approximation (16). The cross mark represents the estimated
flash point position, calculated by using the method described in Sec. V.B.
Fig. 6. Illustration of an off-focus coherent FO scenario with a lens based FPA
coupled to a parabolic reflector. Inset shows a dielectric lens under
consideration.
IV. FLY’S EYE LENS ARRAY
In this section, it is clarified how the proposed CFO
methodology can be applied to FPA based on lens antennas.
The geometry of the problem is sketched in Fig. 6.
By extending the applicability region of the FO method, see
(16), a large format lens based FPA such as the one in Fig. 6 is
divided into several regions. In the middle of each region an
off-focus FO sphere is centred. Around the apex position of
each lens element, a local phase linearization is performed, see
(8), where is chosen. As the result, the PWS of the
reflector,
, is derived at the surrounding of the lens
element. Each plane wave of this spectrum is propagated using
a GO approach to a FO sphere defined inside the lens element
as shown in the inset of Fig. 6, as:
(17)
m-th Off-focus
FO sphere
Focusing QO
component
55 65
(dB)
-40
0
56 59 6257 60
-20
61 63
-10
-30
6458
PO
Off-focus CFO
55 65
[deg]
-200
200
56 59 6257 60
0
61 63
100
-100
6458
-50
-150
50
150
PO
Off-focus CFO
55 65
(dB)
-40
0
56 59 6257 60
-20
61 63
-10
-30
6458
PO
Off-focus CFO
55 65
[deg]
-200
200
56 59 6257 60
0
61 63
100
-100
6458
-50
-150
50
150
PO
Off-focus CFO
Parabolic
Reflector
Off-focus FO
Sphere
Off-focus FO
applicability
region
Lens based FPA
Linearization
applicability
region
Elliptical
lens
Lens FO
Sphere
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(a)
(b)
(c)
(d)
Fig. 7. -component of the electric field along the x-axis. Left panel: magnitude of the electric field evaluated on the focal plane of the parabolic reflector. In its
inset, the 2D reflector focal field and the lens position are shown. Middle and the right panel: the magnitude and phase of the electric field are evaluated on the
focal plane of an elliptical lens, respectively. (a) the reflector is scanning 8 beams, the lens is located at , and the lens diameter is .; (b)
scanning 10 beams, , and ; (c) scanning 40 beams, , and resorting to the off-focus FO approach; (d) same
number of beams scanned and as (c) but and . In all the cases, . Grey, blue and orange regions indicate the applicability region
of FO approximations (16), the one of the coherent FO (12), and the position of the lens in the focal plane of the reflector, respectively.
where is the corresponding point on the lens surface, and
is the length of the corresponding transmitted GO ray between
the lens and FO surface (see inset of Fig. 6).
To derive the PWS of the lens fed by the reflector, the GO
fields in (17) are coherently summed:
. (18)
where is the integration domain which is the entire angular
region subtended by the off-focus FO sphere of the reflector.
When the plane waves impinging on the lens are
characterized by small incident angles, i.e. , the GO
field can be approximated (with a 20% maximum error in the
field magnitude estimation) as follows [13]:
(19)
where the term
indicates the linear and the
coma phase shifts;
represents the flash-point
position, when assuming ;
where is the eccentricity of the
elliptical lens. The condition and the FO limit given
in (16), define the validity region of (19).
To check the validity of the above methodology, let us
consider the same reflector geometry described in the previous
section but including a focal plane array of elliptical lenses. In
Fig. 7, the sub-figures to the left panel represent the field at the
focal plane of the reflector, i.e. the direct field on the top of the
lens based FPA. The corresponding direct field cross-section in
the
plane, including the position of the lens, is shown in
each inset. In the middle and right panels, the magnitude and the
phase of the field at the focal plane of a lens are shown,
respectively. Figs. 8(a)-8(c) consider the same parabola as in
Fig. 2, and a lens with diameter , while for Fig. 7(d)
and . The f-number of the elliptical lens is
312
(dB)
-40
40
4 7 105 8
-20
911
-10
-30
6
PO
Coherent FO
-5 5
(dB)
-10
0
10
50
-4 -1 2-3 0
30
1 3
40
20
4-2
PO
Coherent FO
FO
-5 5
[deg]
-200
-100
0
200
-4 -1 2-3 0
100
1 3
150
50
4-2
-50
-150
PO
Coherent FO
FO
515
(dB)
-40
40
6 9 12710
-20
11 13
-10
-30
814
PO
Coherent FO
-5 5
(dB)
-15
25
-4 -1 2-3 0
5
1 3
15
-5
4-2
PO
Coherent FO
-5 5
[deg]
-200
-100
0
200
-4 -1 2-3 0
100
1 3
150
50
4-2
-50
-150 PO
Coherent FO
35 45
(dB)
-40
0
36 39 4237 40
-20
41 43
-10
-30
4438
PO
Coherent FO
-5 5
(dB)
0
40
-4 -1 2-3 0
20
1 3
30
10
4-2
PO
Coherent FO
-5 5
[deg]
-200
-100
0
200
-4 -1 2-3 0
100
1 3
150
50
4-2
-50
-150 PO
Coherent FO
35 45
(dB)
-40
0
36 39 4237 40
-20
41 43
-10
-30
4438
PO
Coherent FO
-5 5
(dB)
0
40
-4 -1 2-3 0
20
1 3
30
10
4-2
PO
Coherent FO
-5 5
[deg]
-200
-100
0
200
-4 -1 2-3 0
100
1 3
150
50
4-2
-50
-150 PO
Coherent FO
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defined as (see inset of Fig. 6), and in all the four
cases (i.e., the lens is truncated).
In Figs. 7(a) and (b), the lens under analysis is positioned at
the focal plane of a reflector at , the plane wave
angles of incidence are and ,
respectively. The results are compared to those obtained using a
standard PO for the left column and multi-surface PO for the
rest. The multi-surface PO code is based on the formulation
described in [32]. The excellent agreement inside the validity
region of the FO is evident. The radius of the central FO
applicability region for the discussed parabolic reflector is
approximately . To demonstrate the necessity of a
coherent FO representation for the reflector’s focal field, in Fig.
7(a) the lens focal field is also calculated assuming a constant
quadratic phase term in the spectrum of the reflector focal field
(1). From this figure, it is evident that one commits a large error
in analysing the coupling of the lens to the reflector by not
accurately describing the quadratic phase term.
In Fig. 7(c)-(d) scenarios which involve off-centred FO
spheres in the x-direction are considered. The lens under
analysis is positioned in the focal plane of the reflector at
and the reflector is illuminated by a plane wave with
an incident angle of . In Fig. 7(d), the propagation
to the lens FO sphere requires the use of the numerical GO, given
in (18), since
. The agreement with the multi-surface
PO evaluation is very good for all case.
V. WIDE FIELD OF VIEW WITH NON-HOMOGENOUS LENS
ANTENNA ARRAYS
It is well known that the scanning capabilities of reflector
antennas are limited for large off-broadside angles. Focal plane
arrays of homogenous (i.e. identical) horns or lenses have
scanning properties proportional to the size of the beam
illuminating the focal plane. In [17], formulas to derive the field
of view (defined with a 3dB scan loss criterion) were given for
opto-mechanical imaging systems. At low frequencies, the use
of feed clusters has been proposed to enlarge the field of view
[25]–[27]. Here, we investigate, instead, the possibility to
enlarge the field of view by properly designing lens based feeds
(lens dimension, lens surface and lens feed). The concept is
applied to a focal plane array where the elements will be non-
homogenous. The feeds of the lens array are placed over a flat
surface to facilitate a monolithic integration at high frequencies.
For lens elements close to the focus of the reflector, the
quadratic phase in (1) and the comma phase in the associated
reflector CFO spectrum are not significant, and a homogenous
lens array can be used with negligible scan penalty.
For mm- and sub-wavelength systems, the use of large f-
numbers (>1) is common due to their intrinsic larger scanning
property [23]. In these cases, the quadratic phase term is the
dominant source of error for off-focus lenses and the CFO has
a dominating linear phase term. To achieve a conjugate field
matching condition, the lens feeds should be laterally displaced
along the lens focal plane with respect to the lens focus. For
elements even farther away from the centre, the CFO spectrum
contains higher order phase terms. These phase terms lead to a
widening of the beams impinging on the lens array. To improve
the coupling to these distorted fields, one can first enlarge the
lens diameters (amplitude matching) and introduce a non-
rotationally symmetric lens feed. Secondly, the phase of the
distorted CFO spectrum can be matched by reshaping the
surface of the lenses. Fig. 8 schematically describes a possible
composition of an optimum focal plane array. Here, different
regions, filled with different types of lenses, have been
identified.
Fig. 8. A large format monolithically integrated FPA based on lens antennas
with a hybrid architecture. The insets show a zoomed in view of the FPA in
different regions and their geometrical parameters.
As an application case, we consider a scenario compatible
with wide-angle QO systems used in the state-of-the-art compact
imaging systems [8], [16]−[18] where antenna gains of about of
to dBi are needed with about beams.
As the baseline for the design of the FPA, we consider a
silicon elliptical lens ( ) of variable diameter and
coated with a standard quarter wavelength matching layer with
relative permittivity of . The parameters of
the considered reflector coupled lens antenna are listed in Table
1. The far field of linearly -polarized lens feeds is modelled
via a Gaussian beam as follows:
(20)
where , and ;
is a normalization factor; and are chosen in such a way
that the antenna far field matches the CFO spectrum at dB
normalized level. The Gaussian patterned antenna feeds are
placed at the lower focus of each elliptical lens.
Figure 9 shows the field on the reflector focal plane when 0
(i.e. broadside direction), 15.5, 23.5, 34, and 43.75 beams are
scanned. The maximum of the focal field for each considered
scanning position is located inside one of the validity region of
the central, 1st, 2nd and 3rd off-focus FO sphere located at
, , and , respectively. When
the reflector is scanning 15.5 beams, the focal field exhibits
asymmetric sidelobes, due to the comma phase terms as
described in [13], while scanning 23.5, 34 and 43 beams the
first two side lobes and the main lobe of the focal field are
merged, due to higher order phase errors.
In Fig. 10, the scan loss of this incident focal field is shown
(solid grey line). The circle mark represents the number of
beams scanned ()) through the parabolic
reflector before reaching a scan loss of . The value is
obtained by using eq. (3) of [17]. It is worth noting that the
incident scan loss curve (solid grey line) calculated here
Parabolic Reflector
Region 1:
Homogenous
lens array with
identical feeds
Region 2:
Homogenous
lens array with
displaced feeds
Region 4:
Non-homogeneous array with
shaped lens surfaces
Region 3:
Non-homogeneous lens
array
Lens based FPA
Antenna Feeder
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matches the standard formulas, and it is in line with the
approximations available in the literature.
Fig. 9. Normalized electric field in the focal plane of the parabolic reflector
when scanning 0 (broadside), 15.5, 23.5, 34, and 43.75 beams. The shaded
regions represent the FO applicability regions. The edge taper level of the field
intercepted by a lens element is also shown as an example.
Fig. 10. Scan loss of the QO system versus the number of beams scanned, for
the geometry reported in Table 1. The yellow, green, and orange regions
represent the identical (21), displaced (22), and enlarged elements (23) regions,
respectively. The value identified by the grey circle symbol shows the number
of beams scanned with less than 3dB scan loss [17]. The cross marks indicate
the scan loss obtained by using the PO solver of GRASP [33].
In the following subsections, the four FPA regions identified
for optimizing the scanning performance of the reflector system
are described. In the top row of Figs. 11(a)-(d), the magnitude
and phase of the CFO spectrum of the parabolic reflector is
shown for several of the cases in Fig. 9.
A. Region 1: Homogenous Lens Array with Identical Feeds
In this region, the diameter of these lenses is chosen as
which roughly corresponds to the width of the main
beam of the reflector focal field when looking at the broadside
direction. In Fig. 11(a), the CFO spectrum of the lens is
compared to the corresponding one calculated from the antenna
far field, when the lens element is placed at the reflector focus.
It can be noted an excellent matching between the two fields
(middle and bottom rows). As a result, the aperture efficiency
for the central array element is about .
Figure 10 shows the scan loss when an array of homogeneous
lenses with identical centred feeds are considered (solid black
lines). It is worth noting that for this lens array the scan loss
reaches only after scanning beams.
(a) (b)
(c) (d)
Fig. 11. Magnitude and phase of the CFO spectrum of the reflector (top row),
and the lens (middle row). The far field of the lens feed is also shown in the
bottom row. (a) The central element of the homogenous lens array with identical
feeds; (b) the element 15.5 beams from the centre of the homogenous lens array
with displaced feed; (c) the element 23.5 beams from the centre of the non-
homogeneous lens array; (d) the element 43.75 beams from the centre of the
non-homogeneous lens array.
The rapid increase of the loss is due to the phase mismatch
between the CFO spectrum and the antenna far field. This phase
mismatch is mainly due to the quadratic phase of the reflector
focal field. One can calculate the quadratic phase difference
over a lens surface as
, where
and represent the edge positions of the lens element on
the reflector focal plane; is the radius of the reflector FO
sphere. Imposing a maximum of phase difference leads to
a scan loss of . Taking this scan loss value as the limit,
the maximum number of beams scanned by homogenous lens
array (i.e. with identical uniformly spaced feed elements)
defines the limit for this region as follows:
(21)
In Fig. 10, this region is marked with a yellow colour. As
expected, at the edge of this region, the identical element array
exhibits about of scan loss. Within the region identified
by (21), the architecture of the proposed optimum lens based
FPA is also synthesized using identical elements. The scan loss
of this array is also shown in Fig. 10 (blue line).
B. Region 2: Homogenous Lens Array with Displaced Feeds
For elements farther away than
, see (21), the CFO
spectrum exhibits a linear phase as can be seen in Fig. 11(b).
One can conjugate match this phase term by displacing the lens
feeds laterally in their respective lens focal planes. In this
503025 35 40 45-5
(dB)
-40
0
015520
-20
-10
-30
10
-5
-15
-25
-35
Center FO
validity region
1st off-focus
FO region
2nd off-focus
FO region
3rd off-focus
FO region
PO
CFO
Broadside
15.5 Beams
Scanned
34 Beams
Scanned
23.5 Beams
Scanned
43.75 Beams
Scanned Edge
taper
level
Lens
Diameter
503025 35 40 45
Scan Loss (dB)
5
0
015520
4
2
10
1
3
Incident Reflector Focal Field
Homogenous Lens Array with Identical Feeds
Combined non-homogeneous Lens Array
TABLE 1
THE GEOMETRICAL PARAMETERS FOR THE SCAN LOSS EXAMPLE
1
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second region, the diameter of the lenses is kept constant over
the array () since higher order phase terms are still
not relevant.
The optimum position for each antenna feed is determined
by using the CFO spectrum of each lens. Specifically, by
finding a position on the focal plane where the phase of the
PWS is minimum, one can estimate where the maximum of the
focal field is located, i.e. the flash-point position. To do so, an
error function
is defined, where
is the phase of
The flash point position, ,
is then estimated as a position on the focal plane where the sum
of this error function over the whole
set (limited by
maximum subtended angle of the FO sphere) , i.e.
, is minimum. To validate the
methodology discussed here, the example case defined in Fig.
5(a) is considered. The flash point position of the scenario is
estimated at
. This position is shown in Fig.
5(a) with a cross mark. As it can be seen, this method
successfully estimated the flash point position in this wide-
angle scenario.
As shown in Fig. 11(b), for an element 15.5 beams away
from the centre of the reflector focal plane both the magnitude
and phase of the incident field and the antenna far field are well
matched reaching an aperture efficiency of 76%. Fig. 12(a)
summarizes the optimum feed position (indicated in Fig. 8)
using the procedure described above.
The limit of this region is associated to the higher order phase
distortions in the reflector CFO, specifically the comma error.
By using the formula derived in [13], for the estimation of the
comma phase in the PWS of a parabolic reflector, one can
calculate a maximum number of beams scanned by the
displaced feeds reaching at most of scan loss, as follows:
(22)
In Fig. 10, this region is marked with a green colour. Within
this region, the architecture of the proposed optimum lens based
FPA is synthesized using the homogenous lens array with
displaced feeds. The scan loss of this array is shown in Fig. 10
(blue line). As expected, at the edge of the region identified by
(22), this array exhibits about dB of scan loss. The
performance of the homogenous lens array with displaced feeds
is significantly improved with respect to the one with identical
elements (black line).Region 3: Non-Homogeneous Lens Array
For elements farther away than
, see (22), the diameter
of the lens elements should increase to compensate the
widening of the reflector focal field due to the higher order
phase distortions. As shown in Fig. 9, this focal field is
asymmetric in this region. We define a larger rim (i.e. diameter)
for the lenses in this region by finding the contour of the
reflector focal field at a certain level with respect to its
maximum, referred here as lens edge taper level. As an
example, Fig. 9 shows that a lens element close to edge of the
FPA is defined with an edge field taper level of dB. An
automatic procedure is established to define the lens rim for
every element by initially using a dB edge field taper.
However, as mentioned in Sec. III, the FO validity region is also
limited in the vertical direction. Therefore, the considered lens
heights and consequently their diameters are limited. In the
described example scenario, this maximum lens diameter is
. The implemented automatic produce limits the
diameters to this number, and consequently, the obtained edge
taper levels are reduced at the edge of the array. Fig. 12(b)
shows the obtained lens diameters and field edge levels for the
considered scenario. The reported edge taper level is for the
worst case of the 1D cut over the lens surface, e.g. for scanning
in -direction along when . As consequence, the
Gaussian beam waists in (20) will be different now in the two
main planes.
(a) (b)
Fig. 12. The geometrical parameters of the synthesized non-homogeneous lens
array. (a) Gaussian feed parameters (black curves), and feed displacement in
the lens focal plane (red curve). (b) Diameter of the lens elements (black curve),
and edge taper level for each lens for the worst case 1D cut over its surface. The
yellow, green, and orange regions represent the identical (21), displaced (22),
and enlarged elements (23) regions, respectively.
Fig. 11(c) shows, for the lens element located 23.5 beams
away from the centre, that the field match between the lens CFO
and Gaussian feed is very good, both in magnitude and phase.
Figs. 12(a) and (b) summarize the optimum Gaussian feed
parameters and lens diameters and for all regions, respectively.
By using the formula derived in [17], one can calculate the
maximum number of beams scanned in this region with a scan
loss below , as follows:
(23)
In Fig. 10, this region is marked with orange colour. Within this
region, the proposed optimum lens-based array is synthesized
using the design steps described in this subsection. As expected,
at the edge of this region, the array exhibits about of scan
loss.
C. Region 4: Non-Homogeneous Array with Shaped Lens
Surface
For elements farther away than
, see (23), the CFO
spectrum cannot be matched with a translated non-symmetric
Gaussian lens feed. Fig. 11(d) shows a significant difference in
phase distribution between the two, leading to about 5dB scan
loss for this case. To improve this scan loss, one can reshape the
surface of the lens to remove the higher order phase terms on
the lens CFO. Specifically, the difference between the phase of
the elliptical lens CFO spectrum and the translated non-
symmetric Gaussian lens feed, referred to as the hologram
phase, is approximated by a Zernike expansion [34], [35]. The
surface of the elliptical lens is then modified using the
following expression:
(24)
where is the modification of the height of the lens (see Fig.
8); and represent positions on the lens surface; is the
503025 35 40 45
or
0.1
0.5
015520
0.3
10
0.2
0.4
4
1
3
2
0
503025 35 40 45
1
6
015520
3
10
5
2
4
Edge Taper Level (dB)
-19
-16
-10
-13
-7
-4
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Zernike approximation of the hologram phase; and
is the -component of the wave-vector in the lens
material.
In the region outside the one identified by (23), the proposed
optimum lens-based array is synthesized according to the
design steps described in this subsection using enlarged lens
elements with modified elliptical surfaces. The scanning
performance of this array is shown in Fig. 10 (blue line).
As an example case, the surface of a lens element located at
is considered. Firstly, the hologram phase
for this example case is calculated. Secondly, this phase is
represented by a
Zernike expansion, Fig. 13(a). Finally,
the required height modification over the elliptical shape is
calculated using (24), Fig. 13(b). The required modification of
the lens surface is within the specifications given by
commercial silicon micro-machining companies [36]. By
reshaping the surface of this lens element, the system scan loss
is improved from to .
(a) (b)
Fig. 13. Shaping the surface of the lens element scanning beams. (a) The
Zernike expansion of the phase needed to be compensated, i.e. the hologram
phase. (b) The required modification on the lens surface.
D. Validation of the Methodology
In this subsection, the coupling of the described quasi-optical
system calculated using the proposed methodology is compared
to the one obtained by performing a PO analysis that exploits
the reciprocity of the problem and studies it in transmission. In
particular, the field radiated outside the lens antenna is obtained
by using an in-house developed PO formulation similar to the
one described in [37]. Depending on the array element under
study, the lens surface is either elliptical or modified elliptical.
According to the size of the lens element and its distance from
the parabolic reflector, the field is calculated in the lens
radiative near field or in the far field region. This field is then
provided to the PO solver of GRASP [33] as a tabulated source
illuminating the parabolic reflector, to obtain the field radiated
by the entire quasi-optical system. In the proposed CFO
method, the first-order PO diffraction effects are taken into
account; while in GRASP simulation, the diffraction
contribution from the edges (using PTD method) are also
included.
Table 2 compares the aperture efficiency, evaluated with
both methods for the four considered examples in Fig. 11. The
same excellent agreement can be observed in Fig. 14, where the
radiation patterns of the complete quasi-optical system are
shown. Moreover, the scan loss obtained by the PO analysis in
transmission is shown with cross marks in Fig. 10. Again, the
results are very well matched to the ones obtained by the
proposed CFO method. It is worth noting, that the CFO
derivation provides the lens and feed geometries with a single
calculation that lasts about 4 minutes per lens element. In
comparison, the PO analysis in transmission takes about 30
minutes in the same computer. Therefore, this second analysis
procedure would lead to very long elapsed times to find the
optimal lens geometry using iterative simulations. All the
simulations were performed by using a single core Intel i7–
4790 processor with a clock frequency of 3.6 GHz, Cache and
RAM memory of 10MB and 16GB, respectively.
Fig. 14. Radiation pattern of the lens antenna elements coupled to the parabolic
reflector (). These elements are scanning broadside (), 15.5
beams (), 23.5 beams (), and 43.75 beams ().
The solid lines, and dot marks represent the pattern obtained in transmission by
using PO, and reception by using the proposed method, respectively. The
former and the inset, illustrating the pattern in the u-v plane, are calculated by
using GRASP.
VI. CONCLUSION
Imaging systems at millimetre and sub-millimetre
wavelengths are entering a new era with the development of
large format arrays of detectors. A fly’s eye lens array coupled
to absorbers or antennas is a common FPA architecture for such
imagers. Typically, such FPAs are coupled to a quasi-optical
(QO) system involving reflectors. For large QO systems, a full-
wave electromagnetic analysis is not feasible since it is
numerically cumbersome and time-consuming.
In this paper, the original Fourier Optics (FO) procedure has
been extended to derive the spectrum of the incident field on a
reference system centred on antennas located at a large distance
from the focus. The procedure, named here “coherent” FO, has
been used to express the spectrum of the incident field in
realistic cases which include large arrays of lenses within
reflectors focal planes. In particular, the methodology can be
linked to spectral techniques commonly used for arrays, such as
Floquet mode theory, for analysing absorbing mesh grids, and
antenna in reception formalism to analyse the performance of
antenna feeds in reception. By introducing the off-focus FO
approach, the proposed coherent FO representation can be used
-2 20
Normalized Directivity
Pattern (dB)
-40
0
01610 18
-20
-10
-30
12
-5
-15
-25
-35
2 4 6 8 14
GRASP
CFO
-5 30
Normalized Directivity Pattern
(dB)
-40
0
015520
-20
25
-10
-30
10
-5
-15
-25
-35
TABLE 2
THE APERTURE EFFICIENCY OF THE ARRAY ELEMENTS
Sec. V.A
Broadside
Sec. V.B
15.5 beams
Sec. V.C
23.5 beams
Sec. V.D
43.75 beams
Proposed
CFO
method
GRASP
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to analyse, and design systems composed of tens of thousands
of pixels, while the original FO would provide accurate spectra
for only a few tens of lenses. The technique can be used to
assess the scanning performance of large format lens based
FPAs. In particular, by using the developed analysis tool, it was
shown that a scan loss lower than the one of the direct field
(given by standard formulas in the literature) can be achieved
for a wide-angle optics coupled to a lens based FPA. The
proposed array is synthesized according to the described design
rules, namely field matching between the CFO spectrum and
the far field of the lens feed. It is worth noting that in this design
process no numerical optimization algorithms were employed.
Here, scan loss of less than has been achieved while
scanning up to ( beam-widths) for an example
relevant to the state-of-the-art wide-angle imaging systems with
reflector f-number of and directivities of . Finally, the
proposed technique has been validated via a standard Physical
Optics based analysis in transmission with excellent agreement.
APPENDIX: GO PROPAGATION THROUGH DIELECTRIC
MATERIALS
The EM fields reflected by curved multiple surfaces can be
evaluated using a GO formalism as described in [29], [38], [39].
The propagation of GO fields through dielectric surfaces is
instead, to our knowledge, not exhaustively treated in the
literature. This appendix summarizes the formulas describing
the field transmission and propagation, a key aspect for
analysing lenses with the proposed CFO formalism.
In particular, the transmitted GO electric field at an
observation point, , inside a dielectric object (Fig. 1A) can be
expressed as follows:
(A1)
where is the transmission dyad; and
are the perpendicular and parallel transmission coefficients
on the surface, respectively; , and represent the
polarization unit vectors of the incident and transmitted rays,
respectively;
is the incident plane
wave on the dielectric object propagating along
direction;
is the distance between the refraction point, , and observation
point, ; is the propagation constant in the denser medium;
and
are the principal radii of curvature of the transmitted
wave front and can be calculated according to an equation
system as follows:
(A2a)
(A2b)
where is the relative permittivity of the dielectric object, and
is the normal unit vector at the dielectric interface pointing
toward the direction of the impinging wave (Fig. 1A); is the
refraction angle;
, and
are the principal unit directions of
the surface; , and are the principal radii of curvature of the
surface.
Fig. A.1: A 2D sketch of an arbitrary transmitting scenario.
It is worth noting that the expression of the GO transmitted
field, (A1), can be derived by asymptotically evaluating the PO
surface integral at the interface between the two media. The GO
ray contribution corresponds to the stationary phase point of
this PO integral. For further details, the reader is addressed to
[30], where the generalization to an arbitrary astigmatic
incident wave front is also discussed.
REFERENCES
[1] S. Rowe, et al., “A passive terahertz video camera based on lumped
element kinetic inductance detectors,” AIP Review of Scientific
Instruments, 87, 033105–1, Mar. 2016.
[2] M. Calvo, et al., “The NIKA2 instrument, a dual-band kilopixel KID array
for millimetric astronomy,” J. Low Temp. Phys., vol. 184, no. 3, pp. 816–
823, Aug. 2016.
[3] L. Ferrari, et al., “Antenna Coupled MKID Performance Verification at
850 GHz for Large Format Astrophysics Arrays,” IEEE Trans. on
Terahertz Sci. and Tech., vol. 8, no. 1, pp. 127–139, Jan. 2018.
[4] S. O. Dabironezare, et al., “A Dual-Band Focal Plane Array of Kinetic
Inductance Bolometers Based on Frequency-Selective Absorbers,” IEEE
Trans. on Terahertz Sci. and Tech., vol. 8, no. 6, pp. 746–756, Nov. 2018.
[5] G. J. Stacey, et. al., “SWCam: the short wavelength camera for the CCAT
Observatory,” Proc. SPIE 9153, Millimeter, Submillimeter, and Far-
Infrared Detectors and Instrumentation for Astronomy VII, Aug., 2014.
[6] J. J. A Baselmans, et al., “A kilo-pixel imaging system for future space
based far-infrared observatories using microwave kinetic inductance
detectors”, ArXiv e-prints, Sept. 2016.
[7] R. Al Hadi, et al., “A 1 k-Pixel Video Camera for 0.7–1.1 Terahertz
Imaging Applications in 65-nm CMOS”, IEEE Journal of Solid-State
Circuits, vol. 47, no. 12, pp. 2999–3012, Dec. 2012.
[8] D. A. Robertson, et al., “The CONSORTIS 16-channel 340-GHz security
imaging radar,” Proc. SPIE 10634, Passive and Active Millimeter-Wave
Imaging XXI, vol. 10634, 2018.
[9] M. Arias Campo, D. Blanco, S. Bruni, A. Neto, and N. Llombart, “On the
Use of Fly’s Eye Lens Arrays with Leaky Wave Feeds for Wideband
Wireless Communications,” IEEE Trans. on Antennas Propag., , vol. 68,
no. 4, pp. 2480–2493, April 2020.
[10] A. W. Rudge and N. A. Adatia, “Offset-parabolic-reflector antennas: A
review,” Proc. IEEE, vol. 66, no. 12, pp. 1592–1618, Dec. 1978.
[11] M. J. Griffin, J. J. Bock, and W. K. Gear, “Relative performance of filled
and feedhorn-coupled focal-plane architectures,” Appl. Opt., vol. 41, no.
31, pp. 6543–6554, Nov. 2002.
[12] V. Rumsey, “On the design and performance of feeds for correcting
spherical aberration,” IEEE Trans. Antennas Propag., vol.18, no.3,
pp.343–351, May 1970.
[13] N. Llombart, S.O. Dabironezare, G. Carluccio, A. Freni, and A. Neto,
“Reception power pattern of distributed absorbers in focal plane arrays: a
Fourier Optics analysis,” IEEE Trans. on Antennas Propag., vol. 66, no.
11, pp. 5990–6002, Nov. 2018.
[14] A. Nagamune and P. Pathak, “An efficient plane wave spectral analysis
to predict the focal region fields of parabolic reflector antennas for small
and wide angle scanning,” IEEE Trans. on Antenna and Propag. vol. 38,
no. 11, pp. 1746–1756, Nov. 1990.
[15] N. Llombart, B. Blázquez, A. Freni, and A. Neto, “Fourier optics for the
analysis of distributed absorbers under THz focusing systems,” IEEE
Trans. Terahertz Sci. Technol., vol. 5, no. 4, pp. 573–583, July 2015.
air
Denser
dielectric
medium
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12
[16] K. B. Cooper, R. J. Dengler, N. Llombart, B. Thomas, G. Chattopadhyay,
and P. H. Siegel, “THz imaging radar for standoff personnel screening,”
IEEE THz Sci. and Technol., vol. 1, pp. 169–182, Sept. 2011.
[17] E. Gandini, J. Svedin, T. Bryllert, and N. Llombart, “Optomechanical
System Design for Dual-Mode Stand-Off Submillimeter Wavelength
Imagers,” IEEE Trans. Terahertz Sci. Technol., vol. 7, no. 4, pp. 393–403,
July 2017.
[18] S. van Berkel, O. Yurduseven, A. Freni, A. Neto and N. Llombart, “THz
Imaging Using Uncooled Wideband Direct Detection Focal Plane
Arrays,” IEEE Trans. on Terahertz Sci. and Tech., vol. 7, no. 5, pp. 481–
492, Sept. 2017.
[19] V. Galindo-Israel, W. Veruttipong, R. D. Norrod and W. A. Imbriale,
“Scanning properties of large dual-shaped offset and symmetric reflector
antennas,” IEEE Trans. Antennas and Propag., vol. 40, no. 4, pp. 422–
432, Apr. 1992.
[20] E. Gandini, A. Tamminen, A. Luukanen, and N. Llombart, “Wide Field
of View Inversely Magnified Dual-Lens for Near-Field Submillimeter
Wavelength Imagers,” IEEE Trans. Antennas and Propag., vol. 66, no. 2,
pp. 541–549, Feb. 2018.
[21] A. Garcia-Pino, N. Llombart, B. Gonzalez-Valdes and O. Rubinos-Lopez,
“A Bifocal Ellipsoidal Gregorian Reflector System for THz Imaging
Applications,” IEEE Trans. Antennas and Propag., vol. 60, no. 9, pp.
4119–4129, Sept. 2012.
[22] W. P. Craig, C. M. Rappaport, and J. S. Mason, “A high aperture
efficiency, wide-angle scanning offset reflector antenna,” IEEE Trans.
Antennas and Propag., vol. 41, no. 11, pp. 1481–1490, Nov. 1993.
[23] J. Ruze, “Lateral-feed displacement in a paraboloid,” IEEE Trans.
Antennas and Propag., vol. 13, no. 5, pp. 660–665, Sept. 1965.
[24] V. Krichevsky and D. Difonzo, “Optimum beam scanning in offset single
and dual reflector antennas,” IEEE Trans. Antennas and Propag., vol. 33,
no. 2, pp. 179–188, Feb. 1985.
[25] A. W. Rudge and M. J. Withers, “New technique for beam steering with
fixed parabolic reflectors,” Proc. Ins. Elec. Eng., vol. 118, no. 7, pp. 857–
863, July 1971.
[26] V. Galindo-Israel, Shung-Wu Lee and R. Mittra, “Synthesis of a laterally
displaced cluster feed for a reflector antenna with application to multiple
beams and contoured patterns,” APS, pp. 432–435, Stanford, USA, 1977.
[27] C. Hung and G. Chadwick, “Corrected off axis beams for parabloic
reflectors,” APS, pp. 278–281, Seattle, WA, USA, 1979.
[28] Seunghyuk Chang and A. Prata, “The design of classical offset Dragonian
reflector antennas with circular apertures,” IEEE Trans. Antennas and
Propag., vol. 52, no. 1, pp. 12-19, Jan. 2004.
[29] P. Pathak, “High frequency techniques for antenna analysis,” Proceedings
of the IEEE, vol. 80, no. 1, pp. 44–65, Jan. 1992.
[30] S. O. Dabironezare, "Fourier Optics Field Representations for the Design
of Wide Field-of-View Imagers at Sub-millimetre Wavelengths," Ph.D
dissertation, Delft University of Technology, 2020, [Online]. Available:
https://doi.org/10.4233/uuid:23c845e1-9546-4e86-ae77-e0f14272517b.
[31] S. Rao and P. Venezia, "Beam Reconfiguration Using Imaging Reflector
Antennas," IEEE International Symposium on Antennas and Propagation
and USNC-URSI Radio Science Meeting, Atlanta, GA, USA, pp. 1481-
1482, 2019.
[32] S. B. Sorensen and K. Pontoppidan, “Lens Analysis Methods for
Quasioptical Systems,” EuCAP, pp. 1–5, Edinburgh, 2007.
[33] TICRA Tools Software, TICRA, Copenhagen, Denmark.
[34] F. Zernike, “Beugungstheorie des Schneidenverfahrens und Seiner
Verbesserten Form, der Phasenkontrastmethode,” Physica I, 1934.
[35] A. Prata and W. V. T. Rusch, “A quadrature formula for evaluating
Zernike polynomial expansion coefficients (antenna analysis),” Digest on
Antennas and Propagation Society International Symposium, San Jose,
CA, USA, 1989.
[36] Veldlaser, 's-Heerenberg, the Netherlands. Available:
http://www.veldlaser.nl/.
[37] O. Yurduseven, D. Cavallo, A. Neto, G. Carluccio, and M. Albani,
“Parametric analysis of extended hemispherical dielectric lenses fed by a
broadband connected array of leaky-wave slots,” IET Microw., Antennas
Propag., vol. 9, no. 7, pp. 611–617, May 2015.
[38] M. Albani, G. Carluccio, and P. H. Pathak, “Uniform ray description for
the PO scattering by vertices in curved surface with curvilinear edges and
relatively general boundary conditions,” IEEE Trans. Antennas Propag.,
vol. 59, no. 5, pp. 1587–1596, May 2011.
[39] R. G. Kouyoumjian and P. H. Pathak, “A uniform geometrical theory of
diffraction for an edge in a perfectly conducting surface,” Proc. IEEE,
vol. 62, no. 11, pp. 1448–1461, Nov. 1974.
Shahab Oddin Dabironezare (S’11–
M’20) received his B.Sc. degree (cum
laude) in Electrical Engineering-
Communications from the Ferdowsi
University of Mashhad, Mashhad, Iran,
in 2013, M.Sc. and Ph.D. degrees in
Electromagnetics at Terahertz sensing
group, Delft University of Technology,
Delft, the Netherlands, in 2015 and
2020, respectively. He is currently a
postdoctoral researcher at the Department of Microelectronics,
Delft University of Technology, Delft, the Netherlands.
His research interests include, wide band antenna designs for
millimeter and sub-millimeter wave applications, wide field-of-
view imaging systems, Quasi-Optical systems, absorber based
passive cameras, and analytical/numerical techniques in
electromagnetic scattering problems.
Giorgio Carluccio (M’19) was born in
1979 and grew up in Ortelle, Italy. He
received the Laurea degree in
telecommunications engineering and the
Ph.D. degree in information engineering
from the University of Siena, Siena,
Italy, in 2006 and 2010, respectively.
In 2008 and 2009 he was an Invited
Visiting Scholar with the ElectroScience
Laboratory, Department of Electrical
and Computer Engineering, The Ohio State University,
Columbus, Ohio, USA. From 2010 to 2012 and from 2013 to
2014, he was a researcher with the Department of Information
Engineering and Mathematics, University of Siena. From 2012
to 2013, he was a researcher with the Department of Electronics
and Telecommunication, University of Florence, Florence,
Italy. In 2012 and 2013 he was a visiting researcher with the
Department of Microelectronics, Delft University of
Technology (TU Delft), Delft, The Netherlands, where he also
was a researcher from 2014 to 2018. Since 2018 he has been a
RF circuits and antenna scientist with NXP Semiconductors,
Eindhoven, The Netherlands, where he focuses on antenna-in-
package and RF devices for automotive radar applications. His
research interests deal with electromagnetic wave theory,
mainly focused on asymptotic high-frequency techniques for
electromagnetic scattering and propagation; and with modeling
and design of antennas: mainly, dielectric lens antennas,
reflectarray antennas, and THz antennas based on
photoconductive materials.
Dr. Carluccio was the recipient of the 2018 EurAAP
International “Leopold B. Felsen Award for Excellence in
Electrodynamics”, and of the 2010 URSI Commission B Young
Scientist Award at the International Symposium on
Electromagnetic Theory (EMTS 2010), where he also received
the third prize for the Young Scientist Best Paper Award.
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Angelo Freni (S’90–M’91–SM’03)
received the Laurea (Doctors) degree in
electronics engineering from the
University of Florence, Florence, Italy,
in 1987.
Since 1990, he has been with the
Department of Electronic Engineering,
University of Florence, first as an
Assistant Professor and, since 2002, as
an Associate Professor of electromagnetism. From 1995 to
1999, he has also been an Adjunct Professor with the University
of Pisa, Pisa, Italy, and since 2010, a Visiting Professor with the
TU Delft University of Technology, Delft, The Netherlands.
During 1994, he was involved in research with the Engineering
Department, University of Cambridge, Cambridge, U.K.,
concerning the extension and the application of the finite
element method to the electromagnetic scattering from periodic
structures. From 2009 to 2010, he also spent one year as a
Researcher with the TNO Defence, Security and Safety, The
Hague, The Netherlands, where he was involved with the
electromagnetic modeling of kinetic inductance devices and
their coupling with array of slots in the THz range. His research
interests include meteorological radar systems, radiowave
propagation, numerical and asymptotic methods in
electromagnetic scattering and antenna problems,
electromagnetic interaction with moving media, and remote
sensing. In particular, part of his research concerned numerical
techniques based on the integral equation with a focus on
domain-decomposition and fast solution methods.
Andrea Neto (M’00–SM’10–F’16)
received the Laurea degree (summa cum
laude) in electronic engineering from the
University of Florence, Florence, Italy,
in 1994, and the Ph.D. degree in
electromagnetics from the University of
Siena, Siena, Italy, in 2000. Part of his
Ph.D. degree was developed at the
European Space Agency Research and
Technology Center Noordwijk, The Netherlands.
He worked for the Antenna Section at the European Space
Agency Research and Technology Center for over two years.
From 2000 to 2001, he was a Postdoctoral Researcher with the
California Institute of Technology, Pasadena, CA, USA, where
he worked with the Sub-mm Wave Advanced Technology
Group. From 2002 to January 2010, he was a Senior Antenna
Scientist with TNO Defence, Security, and Safety, The Hague,
The Netherlands. In February 2010, he became a Full Professor
of applied electromagnetism with the EEMCS Department,
Technical University of Delft, Delft, The Netherlands, where he
formed and leads the THz Sensing Group. His research interests
include the analysis and design of antennas with an emphasis
on arrays, dielectric lens antennas, wideband antennas, EBG
structures, and THz antennas.
Dr. Neto served as an Associate Editor of the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION
(2008–2013) and IEEE ANTENNAS AND WIRELESS
PROPAGATION LETTERS (2005–2013). He is a member of
the Technical Board of the European School of Antennas and
organizer of the course on antenna imaging techniques. He is a
member of the Steering Committee of the Network of
Excellence NEWFOCUS, dedicated to focusing techniques in
mm and sub-mm wave regimes. In 2011, he was the recipient
of the European Research Council Starting Grant to perform
research on Advanced Antenna Architectures for THz Sensing
Systems. He was the recipient of the H. A. Wheeler Award for
the best applications paper of 2008 in the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
the Best Innovative Paper Prize of the 30th ESA Antenna
Workshop in 2008, and the Best Antenna Theory Paper Prize of
the European Conference on Antennas and Propagation
(EuCAP) in 2010. In 2011, he was the recipient of the European
Research Council Starting Grant to perform research on
advanced antenna architectures for THz sensing systems.
Nuria Llombart (S’06–M’07–SM’13–
F’19) received the Master’s degree in
electrical engineering and Ph.D. degrees
from the Polytechnic University of
Valencia, Valencia, Spain, in 2002 and
2006, respectively.
During her Master’s degree studies,
she spent one year at the Friedrich-
Alexander University of Erlangen-
Nuremberg, Erlangen, Germany, and worked at the Fraunhofer
Institute for Integrated Circuits, Erlangen, Germany. From
2002 to 2007, she was with the Antenna Group, TNO Defense,
Security and Safety Institute, The Hague, The Netherlands,
working as a Ph.D. student and afterwards as a Researcher.
From 2007 to 2010, she was a Postdoctoral Fellow with the
California Institute of Technology, working with the
Submillimeter Wave Advance Technology Group, Jet
Propulsion Laboratory, Pasadena, CA, USA. She was a
“Ramón y Cajal” fellow in the Optics Department,
Complutense University of Madrid, Madrid, Spain, from 2010
to 2012. In September 2012, she joined the THz Sensing Group,
Technical University of Delft, Delft, the Netherlands, where as
of February 2018 she is a Full Professor. She has coauthored
more than 150 journal and international conference
contributions. Her research interests include the analysis and
design of planar antennas, periodic structures, reflector
antennas, lens antennas, and waveguide structures, with
emphasis in the THz range.
Dr. Llombart was the recipient H. A. Wheeler Award for the
Best Applications Paper of 2008 in the IEEE
TRANSACTIONS ON ANTENNAS AND PROPAGATION,
the 2014 THz Science and Technology Best Paper Award of the
IEEE Microwave Theory and Techniques Society, and several
NASA awards. She was also the recipient of the 2014 IEEE
Antenna and Propagation Society Lot Shafai Mid-Career
Distinguished Achievement Award. She serves as a Board
member of the IRMMW-THz International Society. In 2015,
she was the recipient of European Research Council Starting
Grant.