# Near-field Beam Steering with Planar Antenna Array
††thanks: This research was funded by the Slovenian Research And Innovation
Agency (ARIS) grant no. J2-4461, P2-0016, and PR-12348.

Aleš Simončič12, Andrej Hrovat12, Grega Morano12, Teodora Kocevska1, Tomaž
Javornik12  1Department of Communication Systems, Jožef Stefan Institute,
Ljubljana, Slovenia  2Jožef Stefan International Postgraduate School (IPS),
Ljubljana, Slovenia.  Emails: {ales.simoncic, andrej.hrovat, grega.morano,
teodora.kocevska<EMAIL_ADDRESS>
###### Abstract

Beam steering enables manipulation of the electromagnetic radiation patterns
in antenna array systems. A methodology for steering beams in the near field
of a planar antenna array with known phase wavefront functions towards
arbitrary azimuth and elevation angles is described in this paper. Rotation of
the phase wavefront function is used while preserving the shape. The phase
shifts for antenna element excitation currents are determined based on the
distances from antenna elements to the nearest point on the rotated wavefront.
Beam steering utilizing a Gaussian and a Bessel beam is studied. Phase
distribution examples for various steering directions are considered. For
Bessel beam steering, a discussion on the resulting beam shapes and the
steering impact on the polarization mismatch is provided. The results show a
non-negligible magnitude of the cross-polarization with values depending on
the steering direction.

###### Index Terms:

beam steering, Bessel beam, Gaussian beam, near field, phase distribution,
polarization mismatch

##  I Introduction

The deployment of extremely large antenna arrays in high-frequency bands to
support the requirements of future wireless networks implies that wireless
communications systems will operate in the near-field region different from
the conventional systems typically operating in the far-field [1]. The
Fraunhofer distance distinguishes the near field from the far field regions
for an arbitrary antenna, with the boundary between these regions being
proportional to the square of the maximum dimensions of the antenna [2]. In
the context of an antenna array with a large number of antenna elements,
typically, the largest dimension of each antenna element is significantly
smaller than the largest dimension of the entire antenna array. Consequently,
we infer that the far-field condition for each antenna element is much smaller
than the far-field condition of the entire antenna array. This observation
allows us to identify a region where antenna elements operate in the far
field, thereby justifying the far-field approximations for modeling the
emitted electromagnetic waves while the overall antenna array continues to
operate in the near field. This presents an opportunity to exploit the antenna
array’s near field and employ wavefront engineering techniques to generate
beams that would not be achievable in the far field [3].

Through manipulation with the current excitation of the antenna elements in
both magnitude and phase, the generation of various phase wavefronts
representing the line that connects the points of a generated wave with same
phase is possible, leading to the creation of diverse beams. In the
literature, mostly the beam focusing, Bessel beams, Airy beams, and bottle
beams, with the possibility of incorporating orthogonal angular momentum to
the transmitted wave, have been studied [4, 5]. The beams are used in
contemporary applications areas such as energy focusing, blockage mitigation,
and multiple uncorrelated channels.

In practical settings, such as in wireless communication systems or radar
applications, with static antenna arrays and dynamic environments due to the
movement of objects or changes in the propagation medium, the need for beam
steering is emphasized. The use of fixed beams that can not adapt to
environmental changes in real-time is inadequate. The ability to steer beams
arbitrarily is essential for maximizing the performance and adaptability of
antenna arrays.

In this paper, a methodology to steer the beam with known phase wavefront
function in three-dimensional space is described. The polarization mismatch in
the near field resulting from the beam steering process for a Bessel beam is
addressed.

The structure of this paper is as follows. After the introduction, an overview
of previous work is given in Section II, and the system model is presented in
Section III. Section IV presents the method for steering the beams with a
known phase wavefront function. Subsequently, in Section V Gaussian and Bessel
beam steering is studied, along with phase distributions for various steering
directions and presentation of the corresponding figures of steered beams. The
paper concludes with a summary and discussion. For denoting the vectors
boldface is used. The symbol $\mathbf{u}$ represents the unit vectors.

##  II Related work

Significant advancements in wavefront engineering are reported in the
literature, particularly in the domain of optics, where the insights and
methodologies for tailored manipulation of electromagnetic waves enabled many
applications ranging from imaging to laser beam shaping [6, 7, 8]. The use of
the available techniques in radio wave communication and sensing requires
careful modification to consider the longer wavelength, hardware constraints,
interaction with the propagation environment, and application requirements [9,
10].

In [11], Bessel beams were compared with Gaussian beams as representative of
conventional far-field beamforming techniques and the numerical analysis was
reported. The results highlight the advantages of Bessel beams, particularly
in enhancing the resilience of communication links within (sub)THz frequency
bands and near-field environments. The implementation of Gaussian beams and
beam focusing techniques was studied, and the influence of the user position
uncertainty on received signal strength was analysed in [12]. The results show
that beam focusing provides significant gain improvements and is sensitive to
inaccuracies in the user position information attributed to the small focused
area.

The use of wide bandwidth signals affects beam focusing due to the induced
beam-splitting effect and causes alteration of the propagation range for
Bessel beams as discussed in [13]. To mitigate the adverse effects of the wide
bandwidth on miss-focusing without resorting to the true time delays, a method
based on the derivation of a frequency-modulated continuous wave (FMCW) as an
optimal spatial phase modulation function was proposed in [14].

The polarization of the emitted signal from an antenna array has spatial
variability in the near-field, contrary to the far-field scenario where a
single polarization type is typically assumed. Consequently, the received
signal strength fluctuates due to the movement of mobile users in front of the
antenna array, although the polarization of the mobile user is constant. The
impact of polarization mismatch and the potential of its mitigation using
phased arrays with dynamic polarization control was studied in [15]. Contrary
to the spatially invariant optimal configuration observed in far-field
scenarios, spatial variation across the array in the near field was shown.

##  III System model

In this work, we consider a planar antenna array with $M$ antenna elements
located in the $xz$-plane of a rectangular coordinate system, as illustrated
in Figure 1 with yellow rectangles representing the array elements. We denote
$\mathbf{r_{an}}=(x_{an},y_{an},z_{an})$ as the vector from the origin of the
coordinate system to $n$-th antenna element, where $n$ is the index of antenna
element ranging from 1 to $M$. The vector $\mathbf{r}$ extends from the centre
of the coordinate system to the point $P$ in which the electric field is
calculated, while $\mathbf{r_{n}}$ represents the vector from the $n$-th
antenna element to point $P$. The electric field in point $P$ from $n$-th
antenna element is calculated as:

$\mathbf{E_{n}}=\alpha\,I_{n}\frac{e^{-\jmath
k\norm{\mathbf{r_{n}}}}}{\norm{\mathbf{r_{n}}}}\cdot\mathbf{u}_{\theta_{n}},$
(1)

where $\alpha$ is a proportionality constant set to unity given in $V/A$,
$I_{n}$ is the excitation current of the $n$-th antenna element in complex
form, $k$ is the wavenumber, and $\mathbf{u}_{\theta_{n}}$ is the unit vector
defining the polarization of the antenna element in the $\theta$ direction
within the local coordinate system of the antenna element expressed as

$\mathbf{u}_{\theta_{n}}=(\mathbf{u}_{x}\cos{\Phi_{n}}\cos{\theta_{n}}+\mathbf{u}_{y}\sin{\Phi_{n}}\cos{\theta_{n}}-\mathbf{u}_{z}\sin{\theta_{n}}),$
(2)

where $\Phi_{n}$ and $\theta_{n}$ are the azimuth and polar angles in local
coordinates of $n$-th antenna element, respectively and $\mathbf{u}_{x}$,
$\mathbf{u}_{y}$, and $\mathbf{u}_{z}$ are the unit vectors in $x$, $y$, and
$z$ direction, respectively. This model incorporates the polarization mismatch
of emitted electromagnetic waves from antenna elements within the near field
of the antenna array. We assume uniform excitation currents of equal magnitude
and an isotropic radiation pattern for all antenna elements. Furthermore, we
consider the transmitted signals to be monochromatic.

Figure 1: System model. The total electric field at point P contains the
individual contributions of the antenna elements in the $xz$-plane.

##  IV Beam steering

In this section, we describe a method for steering the beams with a known
function of the phase wavefront of the unsteered beam by calculating the phase
shifts of the excitation current of the antenna elements such that the beam is
steered in the arbitrary direction defined by azimuth ($\theta_{Az}$) and
elevation ($\theta_{El}$) as shown in Figure 2(a) for the azimuth angle and
Figure 2(b) for the elevation angle.

Figure 2: Rotation of coordinate system around $z$-axis (a) and $x$-axis (b).

The first step is to determine the transformation matrix $R$ for the rotation
of the original coordinate system by $\theta_{Az}$ around the $z$-axis and by
$\theta_{El}$ around the $x$-axis, as defined by the rotation arrows in Figure
2, so that the $\mathbf{u}_{y^{\prime}}$ direction corresponds to the desired
steering direction. The transformation matrix $R$ is equal to matrix
multiplication of transformation matrix $R_{x}(\theta_{El})$ which rotates the
coordinate system around $x$-axis by $\theta_{El}$ and $R_{z}(\theta_{Az})$
which rotates the coordinate system around $z$-axis by $\theta_{Az}$:

$R=R_{x}(\theta_{El})\cdot R_{z}(\theta_{Az}).$ (3)

$R_{x}(\theta_{El})$ is defined as

$R_{x}(\theta_{El})=\begin{bmatrix}1&0&0\\\
0&\cos{\theta_{El}}&-\sin{\theta_{El}}\\\
0&\sin{\theta_{El}}&\cos{\theta_{El}}\end{bmatrix},$ (4)

and $R_{z}(\theta_{Az})$ is defined as

$R_{z}(\theta_{Az})=\begin{bmatrix}\cos{\theta_{Az}}&\sin{\theta_{Az}}&0\\\
-\sin{\theta_{Az}}&\cos{\theta_{Az}}&0\\\ 0&0&1\end{bmatrix}.$ (5)

The new coordinates are

$\begin{bmatrix}x^{\prime}\\\ y^{\prime}\\\
z^{\prime}\end{bmatrix}=R\cdot\begin{bmatrix}x\\\ y\\\ z\end{bmatrix}.$ (6)

The primary concept underlying the steering mechanism involves utilizing the
wavefront function of an unsteered beam in a rotated or transformed coordinate
system, where the direction of $\mathbf{u}_{y^{\prime}}$ aligns with the
desired steering direction. As the wavefront function retains its shape and
solely undergoes spatial rotation, the resultant beam achieves effective
steering while maintaining its inherent characteristics. Initially, we employ
the unsteered wavefront function in the transformed coordinate system
$y^{\prime}=f(x^{\prime},z^{\prime})$. To express the same wavefront within
the original coordinate system, we utilize the coordinate transformation
equation delineated by (6). Consequently, the obtained phase wavefront
function in the original coordinate system is denoted as
$y_{0}=f(x,z,\theta_{Az},\theta_{El})$.

The subsequent stage involves calculating the minimum distances between each
antenna element positioned in the $xz$-plane and the phase wavefront function
defined in the original coordinate system. This task requires determining the
length of a line originating from the location of the antenna element and
ending at the intersection with the phase wavefront, under the condition that
the line is perpendicular to the phase wavefront, as shown in Figure 3, where
the minimum distance is denoted as $d$ for an antenna element taken as an
example. The equation governing such a line’s characteristics can be expressed
as follows:

$\mathbf{r_{0}}=\mathbf{r_{an}}-t\cdot\mathbf{n},$ (7)

where $\mathbf{r_{0}}$ denotes the coordinates on the surface represented by
the phase wavefronts in the original coordinate system, defined as
$(x,y_{0}(x,z,\theta_{Az},\theta_{El}),z)$, $\mathbf{n}$ represents the vector
perpendicular to the phase wavefront, and $t$ is a constant value proportional
to the length of the line.

The perpendicular vectors $\mathbf{n}$ to the wavefront surface
$y_{0}(x,z,\theta_{Az},\theta_{El})$ are computed as:

$\mathbf{n}=\nabla(y_{0}-y)=(\frac{\partial y_{0}}{\partial
x},-1,\frac{\partial y_{0}}{\partial z}).$ (8)

Figure 3: Minimal distance between an antenna element and the phase wavefront.

Expanding the expression of (7) while taking into account the aforementioned
equalities yields:

$(x,y_{0}(x,z,\theta_{Az},\theta_{El}),z)=(x_{an},y_{an},z_{an})-t\cdot(\frac{\partial
y_{0}}{\partial x},-1,\frac{\partial y_{0}}{\partial z}).$ (9)

The foregoing expression constitutes a system of three equations with three
unknowns ($x$, $z$, $t$). By solving this system, we can ascertain $x_{i}$ and
$z_{i}$, which correspond to the coordinates where the line originating from
the antenna element’s position intersects the phase wavefront. The point of
intersection is $(x_{i},y_{0}(x_{i},z_{i},\theta_{Az},\theta_{El}),z_{i})$.
The distance $d$ is equal to the Euclidean distance between antenna element
coordinates and the point of intersection:

$d_{an}=\sqrt{(x_{i}-x_{an})^{2}+y_{0}(x_{i},z_{i},\theta_{Az},\theta_{El})^{2}+(z_{i}-z_{an})^{2}}.$
(10)

The phase progression induced in the transmitted wave due to the distance $d$
traveled from antenna element $n$ is determined as follows:

$\Delta\Phi_{an}(x_{an},z_{an})=\frac{2\pi d_{an}}{\lambda},$ (11)

where $\lambda$ is the wavelength of the transmitted signal. The excitation
required for each antenna element to counteract the influence of phase
progression as the signal travels towards the phase wavefront, thereby
producing the steered beam, is expressed as $I_{n}=e^{\jmath\Delta\Phi_{an}}$.
A depiction of the phase shifts in excitation corresponding to the $x$ and $z$
coordinates is commonly referred to as the phase distribution.

It is worth noting that the distance $d$ corresponds to the shortest distance
between an antenna element and the phase wavefront. Therefore, an alternative
approach to calculating the distance $d$ involves minimizing the squared
distance function between two points (the antenna element coordinates and the
closest point on the phase wavefront) denoted as

$f_{d}=(x-x_{an})^{2}+y_{0}(x,z,\theta_{Az},\theta_{El})^{2}+(z-z_{an})^{2},$
(12)

by solving for the unknown $x$ and $z$ coordinates through the system of two
equations

$\frac{\partial f_{d}}{\partial x}=0,\,\,\,\frac{\partial f_{d}}{\partial
z}=0.$ (13)

A similar optimization methodology involves the utilization of Lagrange
multipliers. Despite yielding identical results, we observed that the method
involving the computation of normal vectors on the phase wavefront surface is
more efficient. Therefore, we focus our subsequent analysis on this method.

##  V Beam steering examples

This section describes the examples of steering Gaussian and Bessel beams [16]
using the method described above. We begin by describing the steering of
Gaussian beams since the result corresponds to beam steering in the far field,
which is well-studied and serves as a reference for the described steering
method. Next, the steering of the Bessel beam is described as a representative
beam in the near field, also addressing the lack of studies on how the
steering affects the beam shape. The simulations for both beams were performed
with a grid of 100x100 antenna elements operating at a frequency of 100 GHz
with a spacing between the antenna elements of $\lambda/2$.

###  V-A Gaussian beam steering

The phase wavefront of an unsteered Gaussian beam emitted from an antenna
array on $xz$-plane can be described as a plane with equation $y=0$ [3]. The
phase distribution on $xz$-plane generating the unsteered Gaussian beam is
shown in Figure 4(a). The steering of the Gaussian beam in an arbitrary
direction is accomplished by describing the unsteered Gaussian beam function
in the rotated coordinate system that corresponds to the steering direction.
In the rotated coordinate system, the plane of the phase wavefront has the
equation $y^{\prime}=0$. Reverting back to the original coordinate system
using the transformation defined in (6), the equation of the tilted phase
wavefront plane in the desired direction is given by:

$y_{0}=x\,\tan{\theta_{Az}}+z\,\frac{\tan{\theta_{El}}}{\cos{\theta_{Az}}}.$
(14)

Solving the system of equation defined by (9) and utilizing the equation (10),
the minimal distance of the antenna element to the rotated phase wavefront is
computed, along with the corresponding phase shifts (11). The phase shifts
applied to each antenna element, as a function of the antenna element’s
coordinates in the $xz$-plane, are expressed as:

$\Delta\Phi(x_{a},z_{a})=k\,(x_{a}\cos{\theta_{El}}\,\sin{\theta_{Az}}+z_{a}\,\sin{\theta_{El}}),$
(15)

which is a well-established equation for beam steering in the far field [2],
confirming the validity of the described steering approach. Figures 4(b), (c),
and (d) illustrate the phase distributions derived using (15) for various
$\theta_{Az}$ and $\theta_{El}$ values.

Figure 4:  Phase distributions on antenna array surface for generation of
Gaussian beams with varying $\theta_{Az}$ and $\theta_{El}$ angles.

###  V-B Bessel beam steering

The unsteered Bessel beam’s phase wavefront manifests as a cone in the
positive domain of the $y$-axis [11], which is represented mathematically as
follows:

$y=\frac{h}{r}\,\sqrt{(x^{2}+z^{2})},$ (16)

where $r$ is the radius of the cone at high $h$. Steering the Bessel beam
involves rotating the cone by $\theta_{Az}$ in azimuth and $\theta_{El}$ in
elevation while maintaining the cone’s geometry. In the equivalently rotated
coordinate system ($x^{\prime}$, $y^{\prime}$, and $z^{\prime}$), the equation
of the cone is expressed as:

$y^{\prime}=\frac{h}{r}\,\sqrt{(x^{{}^{\prime}2}+z^{{}^{\prime}2})}.$ (17)

By utilizing the transformation defined by (6), we derive the equation of the
rotated cone within the original coordinate system.

Determining the minimum distance between the antenna element and the nearest
point on the rotated phase wavefront involves solving the system of equations
specified by (9) and utilizing equation (10), as also depicted in Figure 3.
The required phase shifts for antenna elements are calculated with (11). As
the analytical solution to the system of equations (9) was not found, the
distances and, consequently, the phase progressions were calculated
numerically in the Matlab software environment. Figure 5 shows the acquired
phase distributions on the $xz$-plane, which represent the phase shifts that
must be applied to the antenna elements to produce the desired beam for
various $\theta_{Az}$ and $\theta_{El}$ values.

Figure 5: Phase distributions on antenna array surface for generation of
Bessel beams with varying $\theta_{Az}$ and $\theta_{El}$ angles.

Figures 6, 7, and 8 show the Bessel beams generated by the phase distributions
5(a), (b), and (c), respectively. The electric field of x-, y- and
z-polarization was computed at each spatial point. We only plotted the
electric field polarizations that do not have negligible electric field
values. For the unsteered beam, only the z-polarization is shown, as the x-
and y-polarizations have negligible values. When the beam is steered, the
polarization mismatch is different when the beam is steered in azimuth and
elevation. With azimuthal steering, only z-polarization is present, while
steering in elevation results in significant y-polarization values. This
presents a challenge since most receivers are designed to receive only one
type of polarization, which reduces the power efficiency of the system. The
reason for this is that we have assumed that the polarization of the antenna
elements is $\mathbf{u}_{\theta_{n}}$, a polarization associated with dipole
antennas. The x-polarization contributions of the antenna elements to the
electric field, calculated on the $xy$-plane or the $yz$-plane, cancel each
other out. Similarly, the y-polarization contributions for points on the
$xy$-plane cancel each other out. For this reason, only the z-polarization is
significant if (1) the beam is not steered, since most of the beam is
concentrated around the $y$-axis, or (2) the beam is steered only in azimuth
since the beam is concentrated on the $xy$-plane. If the beam is steered in
elevation, the contributions of the y-polarizations do not cancel each other
out, so the y-polarization values are not negligible. At points not on the
$xy$\- or $yz$-plane, all polarizations are present, with the influence of x-
and/or y-polarization increasing with distance from the $y$-axis, further
enhancing the effect of polarization mismatch. Consequently, if the beam is
steered in both azimuth and elevation directions, all polarizations are
expected, with the values of y-polarization increasing with increasing
elevation angle and the values of x-polarization increasing with increasing
azimuth angle.

The propagation range of Bessel beams is inherently limited by the finite
transmit power and is a function of the antenna array’s aperture size and the
$h/r$ ratio [11].

Figure 6: Bessel beam without steering. Figure 7: Bessel beam steered in
azimuth, $\theta_{Az}$ = $20^{\circ}$. Figure 8: Bessel beam steered in
elevation, $\theta_{El}$ = $20^{\circ}$.

##  VI Conclusion

This paper studies a method for steering beams with known phase wavefront
functions towards arbitrary azimuth and elevation angles. The method involves
rotation of the phase wavefront function by azimuth and elevation angle while
maintaining the shape and estimation of the distance from the antenna elements
to the nearest point on the rotated wavefront. The distances are proportional
to the required phase shifts for the antenna element’s excitation current to
generate the steered beam.

Two examples of beam steering are presented. The first example utilized a
Gaussian beam, while the second example focused on steering a Bessel beam.
Phase distribution examples for various steering directions are considered.
Additionally, for Bessel beam steering, the resulting beam shape is analysed,
and its impact on polarization mismatch within the antenna array’s near field
is discussed. When the beam is steered only in azimuth there is no
polarization mismatch assuming the receiver with z-polarization. However,
steering the beam in elevation leads to non-negligible values of the cross-
polarization, which has to be taken into account. When azimuth and elevation
steering are used, all three polarizations exhibit non-negligible values, with
varying magnitudes depending on the azimuth and elevation angles of the
steering direction.

In future work, the proposed methodology can be extended to accommodate
arbitrary radiation patterns, signals with higher bandwidth can be considered,
and the potential signal distortions resulting from the steering process can
be identified and studied.

## References

  * [1] Y. Liu, C. Ouyang, Z. Wang, J. Xu, X. Mu, and A. L. Swindlehurst, “Near-Field Communications: A Comprehensive Survey,” Jan. 2024, arXiv:2401.05900 [eess]. [Online]. Available: http://arxiv.org/abs/2401.05900
  * [2] C. A. Balanis, _Antenna Theory: Analysis and Design_. John Wiley & Sons, Dec. 2015.
  * [3] D. Headland, Y. Monnai, D. Abbott, C. Fumeaux, and W. Withayachumnankul, “Tutorial: Terahertz beamforming, from concepts to realizations,” _APL Photonics_ , vol. 3, no. 5, p. 051101, Feb. 2018. [Online]. Available: https://doi.org/10.1063/1.5011063
  * [4] A. Singh, V. Petrov, H. Guerboukha, I. V. A. K. Reddy, E. W. Knightly, D. M. Mittleman, and J. M. Jornet, “Wavefront Engineering: Realizing Efficient Terahertz Band Communications in 6G and Beyond,” May 2023, arXiv:2305.12636 [cs, eess]. [Online]. Available: http://arxiv.org/abs/2305.12636
  * [5] H. Li, X. Wang, S. Wang, W. Sun, J. Ye, P. Han, S. Feng, Y. Yu, and Y. Zhang, “Vector measurement and performance tuning of a terahertz bottle beam,” _Scientific Reports_ , vol. 8, no. 1, p. 13177, Sep. 2018, number: 1 Publisher: Nature Publishing Group. [Online]. Available: https://www.nature.com/articles/s41598-018-31250-7
  * [6] F. Yaras, H. Kang, and L. Onural, “State of the Art in Holographic Displays: A Survey,” _Journal of Display Technology_ , vol. 6, no. 10, pp. 443–454, Oct. 2010. [Online]. Available: http://ieeexplore.ieee.org/document/5473052/
  * [7] P. W. M. Tsang and T.-C. Poon, “Review on the State-of-the-Art Technologies for Acquisition and Display of Digital Holograms,” _IEEE Transactions on Industrial Informatics_ , vol. 12, no. 3, pp. 886–901, Jun. 2016. [Online]. Available: http://ieeexplore.ieee.org/document/7448909/
  * [8] J. Arlt and M. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: The optical bottle beam,” _Optics letters_ , vol. 25, pp. 191–3, Mar. 2000.
  * [9] M. A. Jamshed, A. Nauman, M. A. B. Abbasi, and S. W. Kim, “Antenna Selection and Designing for THz Applications: Suitability and Performance Evaluation: A Survey,” _IEEE Access_ , vol. 8, pp. 113 246–113 261, 2020, conference Name: IEEE Access.
  * [10] R. Pant and L. Malviya, “THz antennas design, developments, challenges, and applications: A review,” _International Journal of Communication Systems_ , vol. 36, no. 8, p. e5474, 2023, _eprint: https://onlinelibrary.wiley.com/doi/pdf/10.1002/dac.5474. [Online]. Available: https://onlinelibrary.wiley.com/doi/abs/10.1002/dac.5474
  * [11] A. Singh, A. J. Alqaraghuli, and J. M. Jornet, “Wavefront Engineering at Terahertz Frequencies Through Intelligent Reflecting Surfaces,” in _2022 IEEE 23rd International Workshop on Signal Processing Advances in Wireless Communication (SPAWC)_ , Jul. 2022, pp. 1–5, iSSN: 1948-3252. [Online]. Available: https://ieeexplore.ieee.org/document/9833975
  * [12] D. Acharya, J. Kokkoniemi, A. Parssinen, and M. Berg, “Performance Comparison of Near-Field Focused and Conventional Phased Antenna Arrays at 140 GHz,” in _2021 Joint European Conference on Networks and Communications & 6G Summit (EuCNC/6G Summit)_. Porto, Portugal: IEEE, Jun. 2021, pp. 1–6. [Online]. Available: https://ieeexplore.ieee.org/document/9482501/
  * [13] A. Singh, V. Petrov, and J. M. Jornet, “Utilization of Bessel Beams in Wideband Sub Terahertz Communication Systems to Mitigate Beamsplit Effects in the Near-field,” in _ICASSP 2023 - 2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_ , Jun. 2023, pp. 1–5.
  * [14] N. J. Myers and R. W. Heath, “InFocus: A Spatial Coding Technique to Mitigate Misfocus in Near-Field LoS Beamforming,” _IEEE Transactions on Wireless Communications_ , vol. 21, no. 4, pp. 2193–2209, Apr. 2022, conference Name: IEEE Transactions on Wireless Communications.
  * [15] N. J. Myers, Y. Aslan, and G. Joseph, “Near-field focusing using phased arrays with dynamic polarization control,” May 2022, arXiv:2205.10243 [cs, eess]. [Online]. Available: http://arxiv.org/abs/2205.10243
  * [16] J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” _JOSA A_ , vol. 4, no. 4, pp. 651–654, Apr. 1987, publisher: Optica Publishing Group. [Online]. Available: https://opg.optica.org/josaa/abstract.cfm?uri=josaa-4-4-651