Andrea Francesco Morabito,
Loreto Di Donato, and Tommaso Isernia
I
n this article, we propose a method based on the
aperture antennas theory to understand the limitations of orbital angular momentum (OAM)
antennas in far-field links. Additional insight
is also given by analyzing the properties
of the operators relating source and farfield distributions for a given order of
the vortex and emphasizing additional
drawbacks. The degrees-of-freedom
(DoF) of the fields associated with
the different orders of the vortices
are also discussed.
LOOKING FOR INCREASED
CAPACITY IN ANTENNA LINKS
Among novel techniques developed for utilizing the radio spectrum with maximum efficiency,
great attention has recently been
devoted to OAM antennas [1]–[20].
Such systems have, in some cases,
been proposed as a means to improve,
almost indefinitely, the channel capacity
in a link among two antennas [2].
The idea is to observe that an antenna could simultaneously generate different
fields. Each field is associated with a different
amount of orbital momentum, i.e., to a different
angular variation of the phase around the target
direction (e.g., e jz, e 2jz, with z denoting the azimuth
angle in the observation domain). By associating different
image licensed by ingram publishing
Orbital Angular
Momentum Antennas
Understanding actual possibilities
through the aperture antennas theory.
Digital Object Identifier 10.1109/MAP.2018.2796445
Date of publication: 21 February 2018
IEEE Antennas & Propagation Magazine
a p r il 2 0 1 8
1045-9243/18©2018IEEE
59
information with each of these patterns, it is possible to realize
an OAM multiplexing [17] and eventually enlarge at will the
channel capacity [2]. It is synonymous with different modes of
a channel, where the modes are associated with a free-space
link rather than a guiding structure. Much interest has been
devoted to this topic, including spectacular public demonstrations for the media [2] as well as contributions to top scientific
journals [3]–[6]. Since then, doubts [7] and objections [8], [10]
have surfaced, emphasizing expected limitations.
In an attempt to contribute to this debate, we propose a
simple, yet instructive point of view on the subject. In particular,
we focus on the possibility of getting a multiplication of channels
[2] in the far-field zone, which is the usual framework for antenna links. In this respect, we focus on aperture antennas. This
assumption does not impair the general validity of discussions
presented in this article; in fact, any antenna can be regarded as
an aperture antenna with the proper choice of an aperture plane
[21]. Also, the arguments presented later in this article could be
rephrased in terms of multipole expansions as well as in terms of
the appropriate radiation integrals. Additional insight is offered
by an analysis of the operators relating source and field behaviors for a given order of the vortex. The outcomes confirm limitations indicated in [7] and [8] and emphasize further drawbacks
of the proposed multiplexing scheme.
UNDERSTANDING LIMITATIONS THROUGH
HANKEL TRANSFORMS
x
ρ′
a1
y
z
θ1
a2
R
FIGURE 1. A simple communication link wherein the
receiving aperture antenna (on the right) is located in the
broadside direction of the transmitting aperture antenna
(on the left).
60
f ^ tl , zlh =
3
/
, =-3
f, ^ tlh e j,zl, (1)
where
f, ^ tlh = 1
2r
2r
f ^ tl , zlh e -j,zl dzl.(2)
#
0
If kl and z denote the radial and azimuth coordinates in the
spectral domain, respectively, then the Fourier transform of the
source (1) can be written as [25]
2r
F ^ kl , zh = 1
2r
3
# #
0
0
f ^ tl , zlh e -jkltlcos^zl -zh tl dtl dzl , (3)
which can be expanded in a multipole series as follows:
F ^ kl , zh =
3
/
, =-3
F, ^ klh e j,z, (4)
where
F, ^ klh = 1
2r
2r
#
0
F ^ kl , zh e -j,z dz. (5)
Finally, by substituting (1) into (3) and then using (4), one
achieves [25]
As the far field of an aperture antenna is proportional (with the
exception of an element factor) to the Fourier transform of the
aperture field [21], a very convenient mathematical setting is
suggested in [22] and [23]. The latter is based on the exploitation
of Hankel transforms, whose properties are extensively analyzed
and discussed in [24] and [25]. Hankel transforms have been
used [12] to understand some of OAM antennas’ limitations.
However, the following analysis is completely different and
provides a number of novelties with respect to [12]. These latter
are based, in fact, either on the kernel of the involved operators
(in the “A Few Straightforward Consequences” section) or on a
singular value decomposition (SVD) (in the “Further Tools and
Analysis” section).
With respect to our specific problem, Hankel transforms can
be exploited as follows. Let f ^ tl , zlh denote the component of
φ′
interest of the aperture field, with tl and zl , respectively, being
the radial and angular variables spanning the aperture, which is
supposed to be circular and located in the xy plane (Figure 1). By
virtue of [25], f ^ tl , zlh can be expanded in a multipole series as
F, ^ klh =
3
#
0
f, ^ tlh J , ^ kl tlh tl dtl = G , " f, ^ tlh,, (6)
where J , is the ,th-order Bessel function of first kind and
expression (6) is the Hankel transform [24] of order , of the
function f, ^ tlh .
The visible part of the spectrum, i.e., kl # b (b = 2r/m
denotes the wavenumber, and m represents the wavelength),
determines the actual far-field behavior. In particular, by
denoting i as the elevat ion angle with respect to the
boresight and adopting the usual correspondence with
the spectral variables, u = b sin i cos z and v = b sin i sin z,
the following is true:
kl = u 2 + v 2 = b sin i.(7)
Equations (1)–(7) imply a number of simple, yet interesting consequences:
1) An angular variation of order , of the source in terms of the
zl variable corresponds to an angular variation of order , of
the far field in terms of the z variable. Hence, a natural diagonalization of the relationship between the aperture field
and the corresponding spectrum occurs.
2) For any fixed order , of angular variation, the function
f, ^ tlh univocally determines the corresponding function
F, ^ klh, and vice versa. Both the forward and backward relations are ruled by a Hankel transform of order ,.
3) The Hankel-transform relationships (6) determine (but
for a slowly varying factor) the far-field power pattern
april 2018
IEEE Antennas & Propagation Magazine
associated to each source component f, ^ tlh e j,zl .
This last point can also be considered in the reverse fashion, i.e.,
once a desired power pattern is
specified in terms of i and z and
some reasonable prolongation is
used for the invisible part of the
spectrum [26], relations unequivocally define the corresponding
source. In particular, if a given
angular variation of the kind e j,z
is desired in conjunction with a given elevation behavior specified by some F, ^ b sin i h function, then the inverse Hankel
transform will allow for the determination of the corresponding source. In fact, after prolonging the F, ^ klh function in the
invisible part of the spectrum, by using the inverse Hankel
transform [25], the sought source will be given (except for the
implicit angular variation) by
the radius of the receiving antenna
and the elevation angle corresponding to its borders (Figure 1):
Authors quantitatively
analyzed the OAM
antennas’ performances
by applying the SVD
tool to the relevant
operators.
f, ^ tlh = H ,-1 " F, ^ klh, =
3
#
0
F, ^ klh J , ^ kl tlh kl dkl .(8)
A FEW STRAIGHTFORWARD CONSEQUENCES
With the goal of gaining some preliminary understanding, we
examine the forward (6) and backward (8) relationships. In
particular, we look at what is going to happen with increased
values of , and limited sizes of the source. As a crucial but
often overlooked circumstance, 6, ! 0 Bessel functions J ,
present a , th order zero in the origin [27]. Moreover, the
J , function is the unique term of (6) depending on kl. Then,
whatever the source at hand, 6, ! 0 the corresponding spectrum (and, hence, the far field) will have a hole (or, better, a
null) in the boresight direction while still being different from
zero elsewhere. Notably, because of the properties of Bessel
functions [see (10)], such a hole will have increased size and
depth with , .
Under these circumstances, and assuming that a receiver
positioned at the broadside direction (Figure 1) is able to
detect and understand the weaker and weaker signal associated with the ,th-order vortex, a significant price is paid. In
fact, the majority of the power radiated by the transmitting
antenna is spread out of the broadside direction, with two
related consequences. First, the field level is largely useless
in a number of spatial directions, potentially violating radiation limits. Second, spectral resources are wasted. In fact, a
number of potential channels based on space reuse (through
multibeam antennas) are occupied by the toroidal power
patterns associated with the vortices. Therefore, unless the
receiving antenna is large enough to intercept the power
maxima, there will be a waste of both power and potential
space-diversity-based channels, as well as an unjustified electromagnetic pollution.
As a further drawback, 6, ! 0 the received signal will
undergo a very rapid decrease with the distance. In fact, if R
denotes the link distance while a 2 and i 1, respectively, denote
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sin i 1 =
a2
.(9a)
R 2 + a 22
Then, if R & a 2 (which is verified in the far-field region), it is
sin i 1 . ^a 2 /R h % 1, (9b)
as a consequence; in the overall
cone 0 # i # i 1, using 9.1.7 of [27], it is
tl b sin i ,
m 6,: 0 1 tl b sin i % , + 1 ,
J , ^ tl b sin i h . 1 c
2
, !
(10)
so that the borders of the antenna correspond to the maximal
power density that can be intercepted.
Notably, (9b) and (10) also state that, in the overall cone
0 # i # i 1, the usual ^1/R h2 power attenuation will be complemented by a ^1/R h2 , factor for the ,th-order vortex field. Such
a characteristic of ,th-order vortices has been previously recognized and emphasized (on the basis of a completely different
argument) in [8] and previously mentioned in [28].
FURTHER TOOLS AND ANALYSIS
Additional insights into the analysis carried out in the “A Few
Straightforward Consequences” section can be achieved by
exploiting the SVD [29] of the relevant operators (6) (as truncated over a finite circular domain). SVD has been already used in
the study of OAM antennas in [7], wherein the authors quantitatively analyzed the OAM antennas’ performances by applying
the SVD tool to the relevant operators.
Herein, the use of SVD is completely different. In fact, we
analyze the behavior of finite-dimensional sources by performing the SVD of the ,th-order relevant operators (6). By so doing,
we will be able to define which kinds of far field can be realized
by exploiting given angularly varying aperture fields, as well as
how much energy is needed for generating angularly varying far
fields having a given , value while possibly guaranteeing given
performances in the broadside direction.
To apply the SVD to (6), it is convenient to rewrite the
latter in the case of a generic source defined over a circular
aperture of limited radius a and to provide some renormalizations of the aperture disk and of the visible region. By so
doing, the following result (apart from inessential constants)
is achieved:
F, ^ k h =
1
#
0
f, ^ t h J , ^ bakt h tdt, (11)
with t = tl/a and k = kl /b (so that both the normalized variables p and k belong to the interval [0, 1]).
Equation (11) leads to the following formulation in terms of
the operator A ,:
61
F, = A , f,, (12a)
where
A , : f, ! L 2 ^0, 1 h " A , f, ! L 2 ^0, 1 h.(12b)
Let us now denote by " v ,, n, v ,, n, u ,, n , the SVD of A ,, i.e.,
the functions and scalars, such that
A , v ,, n = v ,, n u ,, n
A+
, u ,, n = v ,, n v ,, n,
(13)
wherein A , + is the adjoint of A , while v ,, n, v ,, n, and u ,, n,
respectively, denote the nth singular value, right-hand singular
function, and left-hand singular function associated to the , th
OAM mode.
For any fixed size of the source, the SVD (13) can be
computed by applying the theory and formulas given in
“The Singular Values of Ar , ” and “Managing Singularities
of the Singular Functions,” wherein connections with the
generalized prolate spheroidal wave functions discussed in
[22] and [23] are also given.
The singular functions are orthonormal in the spaces of
sources (i.e., t # 1) and far fields (i.e., k # 1), respectively,
so that
1
#
0
v ,, n ^ t h v ),, p ^ t h tdt =
1
#
0
u ,, n ^ k h u ,), p ^ k h kdk = d n, p 6,,(14)
wherein d denotes the Kronecker delta function and )
means complex conjugation. Hence, they can act as a basis
in the respective domain. They also allow a diagonalization
of the relationship between sources and corresponding far
fields. The aperture field and the visible spectrum are then
expressed as
f, ^ t h =
F, ^ k h =
/ a ,,n v ,,n ^th,(15a)
3
n =1
3
/ b ,,n u ,,n ^kh,(15b)
n =1
The Singular Values of Ar ,
Here we aim to derive the expressions of the singular functions
and the singular values of the operator A , . To diagonalize A , in a
simple and efficient fashion, it is convenient (see [22] and [23]) to
introduce the auxiliary operator Ar , , defined by
1
Ar , f, =
#
0
and then using
ur ,, n
, (S3a)
k
vr ,, n
v ,, n =
, (S3b)
u ,, n =
t
f, ^ t h J , ^ bakt h bakt dt, (S1a)
v ,, n
vr ,, n
=
. (S3c)
ba
with
Ar , : f, ! L 2 ^0,1h " Ar , f, ! L 2 ^0,1h . (S1b)
In fact, it will be
1
This operator is equal to the one extensively studied in [22]
through an eigenvalue decomposition (wherein the eigenfunctions
represent the generalized prolate spheroidal wave functions).
Two crucial properties of Ar , are as follows. 1) As it is a compact
A , v ,, n =
self-adjoint operator, its eigenvalues are real [37], and 2) as it is a
normal operator, its singular values are equal to the amplitude of
its eigenvalues [38]. Therefore, the singular values of Ar , results
are equal (except for a change of sign for negative values of the
eigenvalues) to its eigenvalues, which have been analytically
derived in [22]. Conversely, it is very difficult to extract from [22]
the properties of interest herein, so it makes sense to provide an
alternative easier approach.
In this respect, for any fixed size of the source, the singular values
and singular functions of A , can be computed through numerical
discretization by first computing the singular value decomposition
(SVD) of Ar , , i.e., the functions and scalars vr ,, n, vr ,, n, and ur ,, n, such that
=
Ar , vr ,, n = vr ,, n ur ,, n
rA ,+ ur ,, n = vr ,, n vr ,, n, 62
(S2)
#
0
=
J , ^ bakt h
1
bak
1
bak
1
#
0
vr ,, n
t
tdt
J , ^ bakt h bakt vr ,, n dt =
vr ,, n ur ,, n = v ,, n
ur ,, n
= v ,, n u ,, n .
k
1
bak
Ar , vr ,, n (S4)
In summation, to understand the typical behavior of singular
values and singular functions of the different A , operators, the
following steps are necessary:
■■ step 1: the SVD of the auxiliary Ar , operators
■■ step 2: some robust and accurate implementation of (S3a)
and (S3b).
Step 1 does not require any particular care (except for a
sufficiently large number of discretization points). Step 2 requires
instead some trick to avoid the difficulties related to the presence
of singularities at the denominator. Such a problem can, however
be circumvented by using the simple method described in
“Managing Singularities of the Singular Functions.”
april 2018
IEEE Antennas & Propagation Magazine
where a suitable truncation of expansions, which affects the rate of variation of the aperture field along the
radial coordinate, will be used in
actual instances. In particular (see
the “The Forward Problem: Singular
Values and Singular Functions of the
Different Radiation Operators A ,”
section), truncation will be needed to
regularize the inversion procedure
from the desired far field to the aperture field to circumvent superdirectivity problems [30]–[32].
Because of the theory in “The
Singular Values of Ar ,,” section,
the b ,,n coefficients are simply related to the a ,,n coefficients
as follows:
Values and Singular Functions of
the Different Radiation Operators
A ,, ” we show and comment on the
singular values and singular functions associated with OAM sources
generating different OAM modes,
while, in the “Synthesis of the
Aperture Field Required to Generate a Vortex Field Having a Given
Intensity at the Receiver” section,
we present the results achieved by
solving the synthesis problem mentioned at the end of the “Further
Tools and Analysis” section for different values of ,, a 1 = a 2, and R.
In all cases, at least 60 points per wavelength have been used in
both the spatial and spectral domains to discretize the radiation
operators and computing the corresponding singular values and
singular functions.
As fast oscillations
imply a large content
in the invisible part
of the spectrum and
a very high Q -factor
accordingly, such a
source would also have
a very narrow band.
b ,, n = v ,, n a ,, n .(15c)
The singular values can be thought of as scalar (gain) factors
by which each source is multiplied to give the corresponding far
field. Notably, (15a)–(15c) provides two different (related) ways
for understanding some important OAM antennas’ limitations.
First, the behavior of the singular values and singular functions associated with a given source size and different vortex
orders is observed. Second, an ad hoc synthesis problem can
be solved with reference to the simple communication link
depicted in Figure 1, wherein a receiving aperture antenna (see
the “Understanding Limitations Through Hankel Transforms”
section for the definition of the parameters a 2, R, and i 1) is
located in front of a transmitting OAM antenna having a radius
equal to a 1 . In particular, we are interested in understanding
how much power is required on the aperture for getting an
equal power at the receiving points (for identical collecting
areas). Although contributions relying on Hankel transform
already exist, e.g., [12], it is indeed the first time that diagonalization (15a) and (15b) is utilized to investigate the OAM antennas’ actual potentialities.
ANALYSIS AND SYNTHESIS OF OAM ANTENNAS
By utilizing the previously mentioned tools, we examine the
typical outcomes experienced in the analysis and synthesis of
OAM antennas. In the section “The Forward Problem: Singular
THE FORWARD PROBLEM: SINGULAR VALUES AND SINGULAR
FUNCTIONS OF THE DIFFERENT RADIATION OPERATORS A ,
We computed the singular values and the singular functions
of the operator A , for different values of , and for different
sizes of the aperture. Due to the properties of the relevant
operators, the left singular functions (corresponding to the
sources) and the right singular functions (corresponding to
the radiated far fields) behave similarly. As a typical behavior,
the singular values corresponding to the cases a 1 = 4m and
a 1 = 10m are shown for , = 0, 1, 3, 5, and 7 in Figure 2(a)
and (b), respectively.
In agreement with the theory in [33] and [34], for any fixed
value of ,, the singular values exhibit a step-like behavior,
with an exponential decay after a given value of the index n
(e.g., N ,). Then, if , and n are such that n 2 N ,, realization
of the field u ,,n implies, due to (15c), a very strong increase
of the coefficient a ,,n (and, hence, of the energy of the corresponding source). Moreover, whenever n 2 N ,, the corresponding source is extremely fast oscillating, thus leading
to a high-energy, fast-oscillating (and, hence, difficult to be
realized) source behavior. As fast oscillations imply a large
content in the invisible part of the spectrum and a very high
Q-factor [31] accordingly, such a source would also have a very
narrow band.
MANAGING SINGULARITIES OF THE SINGULAR FUNCTIONS
Here we describe a simple method to avoid the difficulties related
to the presence of singularities at the denominator of (S3a) and
(S3b). The divisions required by (S3a) and (S3b) can be numerically
performed in an accurate and reliable fashion by means of the
following three-step procedure.
■■ Step 1: Multiply the functions u
r and vr by k 1/2 and p 1/2,
respectively.
IEEE Antennas & Propagation Magazine
april 2018
Step 2: Perform a polynomial fitting of the functions derived
from step 1.
■■ Step 3: Erase the constant term of the polynomial coming out
from step 2, and lower by one the order of all of the other terms.
In fact, since the functions coming out from step 1 have a zero
in the origin (and, hence, cannot contain a constant term), step 3
is equivalent to respectively dividing them by k and t.
■■
63
theoretical arguments (see below) suggest that the values of N ,
obey the following rule:
,
,
N , = 1 ^ ba - , h = 2 a = N0 .(16)
r
0
0
–10
–10
–20
–20
–30
–40
–50
–60
,=0
,=1
,=3
,=5
,=7
–70
–80
–90
–100
5
r
,=0
,=1
,=3
,=5
,=7
–30
–40
–50
–60
–70
–90
10
15
20
–100
25
2
4
6
8
10
n
n
(a)
(b)
12
0
–5
0.2
–10
n=6
–0.2
n = 25
–0.4
|u,,5|2 (dB)
0.4
0
ν3,n (dB)
r
–80
0.6
–25
–0.8
–35
0.2
0.4
0.6
0.8
1
,=1
,=3
,=5
,=7
–20
–30
0
14
–15
–0.6
–1
m
In fact, the first addendum is equal to a/ ^m/2 h, which is
known to be the number of DoF associated with a circularly
symmetric source of radius a [36]. Furthermore, as previously
discussed, the circumstance that the field must have an ,th
-order zero in the origin suggests that N , must decrease with , .
Both of the results in Figure 2 (and many others) as well as an a
|σ,,n /maxn (σ,,n )|2 (dB)
|σ,,n /maxn (σ,,n )|2 (dB)
In Figure 2(a), N , is smaller and smaller for increasing values of , . Such a circumstance implies that the larger the , ,
the lower the number of linearly independent patterns which
can be realized by finite-energy sources. This is to be expected,
as increased , values mean the fields have to accommodate
wider and wider holes in front of the transmitting antennas,
e.g., Figure 2(d), while necessarily being bandlimited elsewhere
[35]. We also note that all of this analysis is fully consistent
with the general theory of the DoF of the fields radiated by
finite-dimensional sources given in [35]. Our numerical experiments (performed for many possible sizes of the source) and
–40
0
5
ρ
(c)
10
15
θ (°)
(d)
20
25
30
FIGURE 2. The singular values associated the radiation operator for different values of , and n. (a) a 1 = 10m and (b) a 1 = 4m. A
comparison for different n values of the right-hand singular functions corresponding to (c) , = 3 and a 1 = 4m. A comparison
for different , values of the left-hand singular functions corresponding to (d) n = 5 and a 1 = 4m.
64
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IEEE Antennas & Propagation Magazine
posteriori check (see the following
text) suggest that 1/r is the correct
coefficient for such a decrease.
The rule (16) has two crucial
consequences. First, the maximumorder vortex, which can be excited
by a nonsuperdirective source, i.e.,
the maximum feasible value of ,, is
Fields exhibiting
azimuthal variations of
arbitrarily large order
are de facto unfeasible
when using finitedimensional sources.
, MAX = 2ra m, (17)
which is achieved by enforcing
N , MAX = 0.
Second, the source’s overall number of DoF, which is equal
to the sum of all feasible N , values, results in the following:
N TOT =
/
N,
, # , MAX
, MAX
= N0 + 2 / N, = N0 + 2
, =1
= 2a + 2 2a 2ra - 2
m
m
m
energy required to generate a far
field having a given , th order and
be able, at the same time, to guarantee a given signal intensity at
the edge of the receiving antenna.
As the fields received (at the borders) can be properly recombined
to maximize the received signal,
solution of the proposed synthesis problem also will allow for
comparing the efforts required to
have an identical received power
through equal collecting areas.
Because of the presence of larger and larger holes in the spectrum for 0 # i # i 1 and , ! 0, consideration of the power
collected over the whole receiver’s area (rather than F, ^i 1 h h
would imply a worsening of the actual high-order vortices’ performances. Therefore, Figures 3 and 4 provide an optimistic
estimation of the capabilities of OAM antennas.
In the different trials, and to avoid superdirective sources
(see previous text), we used a truncation of (15a) and (15b) by
adopting an index slightly larger than N , . Then, we solved
problem (20) for , = 1, , = 2, and , = 3 and for different values
of a 1 = a 2 . Concerning the value of R, we limited the analysis to distances not much larger than the Fraunhofer value
R F = 8a 12 /m. This choice has been induced by the circumstance that the OAM antennas’ performance, in terms of
ratio between received and transmitted powers, decays as
R -2^ , +1h (see [16, eq. (20)] and also [7], [8], [10], and [12]).
For each combination of ,, a 1 = a 2, N ,, and R, to evaluate
the energy necessary to generate a particular vortex order,
the achieved PAP value has been compared with that pertaining to the zero-order field (let us say, PAP0).
2ra m
/
, =1
2ra m
r
/
, =1
` 2a - , j m
r
2
, = 4ra2 .
m
(18)
This number is identical to the one given by formula (26) of
[36], i.e.,
2
N TOT = Area of the2source = 4r ` a j , (19)
m
^m /2 h
which also provides an a posteriori check of (16). Notably, all of these considerations go against the suppositions [2], [9], [11] of arbitrarily enlarging the channel
capacity. Fields exhibiting azimuthal variations of arbitrarily large order are de facto unfeasible when using
finite-dimensional sources.
SYNTHESIS OF THE APERTURE FIELD REQUIRED TO GENERATE
A VORTEX FIELD
80
To understand actual possibilities of multiplexing through vortices, we have solved, for different values of , ! 0, the following
convex programing problem in the unknowns a ,,1, f, a ,,N:
#
0
f, ^ t h td t, (20a)
2
Subject to: Re 6F, ^i 1 h@ $ 1, (20b)
60
PAP/PAP0 (dB)
Minimize: PAP ^a ,, 1, ..., a ,, N h =
1
,=1
,=2
,=3
70
50
40
30
20
and
Im 6F, ^i 1 h@ = 0, (20c)
where the functional in (20a), which is a real and positive
quadratic form of the unknowns, is equal to the aperture
power associated to transmitting antenna (except for a 2r factor), while the linear constraints, (20b) and (20c), guarantee a
spectrum amplitude of at least 0 dB for i = i 1 . Therefore, the
goal of the synthesis problem, (20), is to identify the minimum
IEEE Antennas & Propagation Magazine
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10
0
1
1.5
2
2.5
3
R/RFraunhofer
3.5
4
FIGURE 3. The aperture power required to generate
different OAM states as a function of the link distance (for
a 1 = a 2 = 5m).
65
Two kinds of analysis were carried out. As a first set of numerical trials, we fixed a 1 = a 2 = 5m
and repeatedly solved problem
(20) for R ! 6R F, 4R F@ . The results
achieved are summarized in Figure 3, wherein the ratio between
the aperture powers PAP and PAP0
is plotted as a function of the ratio
between the actual and Fraunhofer
link distances. Two important statements can be made.
1) For a fixed link distance, a small
increase of , corresponds to a
huge increase of the required
aperture power.
2) Whatever the , value, the energy required to generate a
given OAM state is a monotonically increasing function of
the link distance. For R . 4R F, even if , is low, the curves
already achieve extremely large values, e.g., the power re­­
quired to generate an angular variation of order , = 3 is
about 80 dB higher than the one needed in case , = 0.
These results confirm that it is preferable to avoid the
adoption of OAM antennas in long-range communication
links, and it is not possible to realize arbitrarily large values
of , (i.e., arbitrarily large channel capacities). This approach
allows a quantitative evaluation of the price to be paid. All
of these considerations are consistent with the ones made
in [7] and [8] as well as with the outcomes of the few experimental proofs previously provided. Additionally, the results
in Figure 3 (green curve) appear to be consistent with the
results in [20]. In the latter article, the authors simultaneously employ an amplifier on the , =! 1 antennas and an
attenuator on the , = 0 antenna
(for a total of 24 dB) to equalize
the received power.
As a second set of numerical trials, by using a 1 = a 2 and R = R F,
we repeatedly solved problem (20)
for 2a 1 ! ^1m, 15m h . The results
achieved are summarized in Figure 4, wherein the ratio between
the aperture p o w e r s PAP and
PAP0 is plotted as a function of
the antennas’ size. The synthesis outcomes reveal two important circumstances.
1) Whatever the antennas’ size, a
huge increase of the aperture
power is required to realize a small increase of ,
2) Whatever the , value, the input power is a monotonically
decreasing function of the antenna size. This is due to the
fact that increasing a 2 ^a 1 h allows dealing with larger i 1
values, thus mitigating the effects of the holes present in the
field pattern around the boresight direction.
All of these results, based on a rigorous quantitative setting,
confirm the OAM antennas’ limitations mentioned in the
“Understanding Limitations Through Hankel Transforms” and
“A Few Straightforward Consequences” sections and agree with
the results in [7], [8], and [12].
By using an extended
receiving antenna in the
near-field zone, it may
be possible to increase
the channel capacity
with respect to where
the same antenna is in
the far-field region.
50
,=1
,=2
,=3
45
40
PAP/PAP0 (dB)
35
30
25
20
15
10
5
0
2
4
6
8
10
12
Antenna’s Diameter (λ)
14
FIGURE 4. The aperture power required to generate
different OAM states as a function of the antenna size (for
a 1 = a 2 and R = R F).
66
CONCLUSIONS
The classic aperture antennas theory has been used as a point
of view on the actual capabilities of OAM antennas. Such an
approach suggests the use of Hankel transforms as a convenient
and simple tool for understanding the ultimate limitations of
these radiating systems. The adopted theoretical framework,
equipped with the SVD analysis of the radiation operators at
hand, indicates that the adoption of OAM antennas is not convenient in very-long-range communication links. Moreover, the
proposed tools allow for identifying the OAM systems’ number
of DoF and to quantitatively understand how performances of
these antennas vary with order, varying sizes, and distances. The
overall outcomes indicate that, for a fixed size of the source, a
substantial price is paid for generating vortices corresponding to
higher and higher values of , and that an arbitrarily large multiplication of channels in the far-field zone is de facto unfeasible.
The results achieved do not eliminate the possibility of
fruitfully adopting OAM antennas in near-field links. In fact,
by using an extended receiving antenna in the near-field zone,
it may be possible to increase the channel capacity with respect
to where the same antenna is in the far-field region. On the
other hand, for a really meaningful assessment, the realized
number of channels should be then compared with alternative multiple-input, multiple-output/multibeam possibilities by
eventually partitioning the receiving and/or transmitting areas
into smaller antennas. In such a study, the previously discussed
tools would still be of interest. In fact, according to the aperture antenna theory, the evaluation of the field at any distance
april 2018
IEEE Antennas & Propagation Magazine
from the aperture can be carried out by just multiplying the
aperture field’s spectrum by a proper exponential function and
performing an inverse transform [21].
AUTHOR INFORMATION
Andrea Francesco Morabito (andrea.morabito@unirc.it) is an
assistant professor of electromagnetic fields at the Mediterranea
University of Reggio Calabria, Italy. His research is focused on
electromagnetic forward and inverse scattering problems as well
as on antenna theory, design, and optimal synthesis.
Loreto Di Donato (loreto.didonato@ dieei.unict.it) is an
assistant professor of electromagnetic fields with the Department of Electrical, Electronics, and Computer Engineering at
the University of Catania, Italy. His research activities are concerned with inverse scattering problems and microwave imaging
as well as antenna synthesis problems.
Tommaso Isernia (tommaso.isernia@unirc.it) is a full professor of electromagnetic fields at the Mediterranea University
of Reggio Calabria, Italy. His research interests cover inverse
problems in electromagnetics emphasizing phase retrieval,
inverse scattering, and antenna synthesis problems. He is a
Senior Member of the IEEE.
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