\chapter{Phased antenna array basics}

This chapter gives a brief introduction to phased antenna array techniques.
First, the basic theory of phased array antennas is discussed, and then a short
overview of different techniques on how to feed the antennas is given.

%-------------------------------------------------------------------------------

\section{Construction}

As the name suggests, an array antenna is constructed from an array of so-called
array elements. \autoref{fig:lin_array} shows different kinds of how the array
elements can be arranged.

\begin{figure}[ht]
  \centering
  \subfigure[linear]{\includegraphics{bilder/lin_array.pdf}}
  \hspace{1cm}
  \subfigure[planar]{\includegraphics{bilder/plan_array.pdf}}
  \hspace{1cm}
  \subfigure[circular]{\includegraphics{bilder/circ_array.pdf}}
  \caption{Types of phased arrays}
  \label{fig:lin_array}
\end{figure}

The feed for each array element can be realised in two different configurations:
\begin{itemize}[style=multiline]
\item each element is fed with the same signal (same magnitude and phase). This
improves the directivity as well as the gain \cite{gustrau}.
\item the elements can be fed with a phase-shifted signal (but all with the
same magnitude). This is then called a \emph{phased array}. By varying the
phase shift, this configuration allows to move the direction of the main lobe.
\end{itemize}
If the phase shift between the array elements is not fixed but adjustable, it is
possible to measure the angle of incidence of a signal received by the array
\cite{Liou2009}, or to electronically adjust the main lobe of the array when
transmitting \cite{Damman2012, Helaly1990}.

%-------------------------------------------------------------------------------

\section{Principle of operation}

The principle of operation of a phased array is first explained for the receiver
case. Due to the reciprocity property \cite{Elliott, Kraus} of the antennas, the
transmit case is equivalent.

\autoref{fig:lin_array} shows a simple linear antenna array with an incident
plane wave. Plane waves do not exist in reality, but far away from the
transmitting antenna, the spherical waves produced by the antenna can be
approximately viewed as plane waves.

\begin{figure}[ht]
  \centering
  \includegraphics{bilder/array_wavefront.pdf}
  \caption{Schematic overview of a linear antenna array}
  \label{fig:lin_array}
\end{figure}

Assume the plane wave is incident to the array at the angle $\alpha$. Then, for
the array elements on the left side, the wave front has to travel the additional
distance
\begin{equation}
\Delta l = l_1 - l_2 = d \cdot \sin \alpha
\end{equation}
for which the time
\begin{equation}
\Delta t = \frac{\Delta l}{c}
\end{equation}
is needed. This gives a phase shift of
\begin{equation}
\varphi' = \frac{ \Delta t }{ T } = f\,\Delta t = \frac{f\,d\,\sin\alpha}{c}
\end{equation}
with
\begin{equation}
\frac{f}{c} = \frac{1}{\lambda}
\end{equation}
we thus have:
\begin{equation}
\varphi' = \frac{d\,\sin\alpha}{\lambda} \label{eq:arrayphaseshift1}
\end{equation}
Now assume the array elements are spaced at a distance $d = \lambda$ and the
wave is incident at an angle of $\alpha = 90$°. From these conditions,
\autoref{eq:arrayphaseshift1} would yield a phase shift of 1, but actually it
should be $2\,\pi$, so a factor of $2\,\pi$ needs to be included in
\autoref{eq:arrayphaseshift1} from which we get the phase shift relation
\begin{equation}
\varphi = \frac{2\,\pi\,d\,\sin\alpha}{\lambda}
\end{equation}
and with the wavenumber
\begin{equation}
k = \frac{2\,\pi}{\lambda}
\end{equation}
the phase shift can also be expressed as follows:
\begin{equation}
\varphi = k\,d\,\sin\alpha \label{eq:arrayphaseshift2}
\end{equation}
In the transmitting case, it is clear now due to the principle of reciprocity
that the direction in which the waves propagate away from the antenna can be
adjusted by phase shifting the feed signals for the individual array elements
according to \autoref{eq:arrayphaseshift2}. If the phase shift between the
array element feed signals is $\varphi$, the angle will be:
\begin{equation}
\alpha = \arcsin\frac{\varphi}{k\,d}
\end{equation}

As can be seen from this theory, a linear antenna array allows to adjust the
transmitting and receiving angle in one dimension. As a consequence, the angle
can be adjusted in two dimensions if a two-dimensional array of antennas is
used. Such a configuration allows to transmit (or receive) in any direction in
the half space. This is also referenced to as spatial filtering.

\italictitle{Remark}
For the analysis of the linear phased array in the transmitting case, the
Huygens-Fresnel principle normally used for refraction problems in the optics
regime can also be used as a model to explain the angle of the produced wave
front \cite{optics}. Refer to \autoref{fig:refraction}. This
figure\footnote{Image taken from \cite{optics}} shows a wave incident to an air-
water interface. Each point where the wave fronts of the incident wave strike
the interface, a source of a spherical wave can be imagined. These waves
propagate into the water domain and the superposition of the many spherical wave
fronts produces the resulting plane wave.

\begin{figure}[ht]
  \centering
  \includegraphics[width=5cm]{bilder/refraction.png}
  \caption{Refraction of a wave at an air-water interface}
  \label{fig:refraction}
\end{figure}

The arrangement of these small sources of spherical waves at the air-water
interface is very similar to a linear phased array.

%-------------------------------------------------------------------------------

\section{Model for the OAM state}

\autoref{fig:lin_to_circ}(a) shows an example of a phased array with 5 elements.
Assume that the elements are fed with a signal which is phase-shifted by a
constant amount from element to element. This will produce a phase front as
indicated in red.

Now imagine that the two array ends -- the elements \#0 and \#4 -- are folded
together on a circle, as shown in \autoref{fig:lin_to_circ}(b). The phase front
must still have the same phase shift from element to element, but since the
elements are arranged on a circular shape, the effect could be compared to the
effect of a spiral phase plate with an incident plane wave. The SPP was
described in detail in the former project.

\begin{figure}[ht]
  \centering
  \subfigure[linear array]{\includegraphics{bilder/linear_phasefront.pdf}}
  \hspace{1cm}
  \subfigure[circular array]{\includegraphics{bilder/circ_phasefront.pdf}}
  \caption{Converting a linear to a circular array}
  \label{fig:lin_to_circ}
\end{figure}

Between the lase element (\#4 in this case) and the first element (\#0), there
is a phase front dislocation, as needed for the OAM. This also looks very
similar to the SPP.

A null is expected in the centre of the circular array because the phase is
ambiguous.

If the phase shift of the feed signal is increased, the phase dislocation will
also increase and if the sign of the phase shift is inverted, the helicity of
the resulting OAM state is also inverted.

Thus, if one manages to electronically control the phase shift, it would be
possible to generate different OAM states with the same apparatus.

%-------------------------------------------------------------------------------

\subsection{Spherical wave model}
The complex amplitude electric field of a plane wave is given by
\begin{equation}
E = A\,\E^{-\j\,k\,z}   \label{eq:planewave}
\end{equation}
if the wave is assumed to travel into positive $z$ direction.

Now we replace $z$ by
\begin{equation}
z \mapsto \sqrt{x^2 + y^2 + z^2}
\end{equation}
which yields
\begin{equation}
E = A \exp\left(-\j\,k\sqrt{x^2+y^2+z^2}\right) \label{eq:simplesphericalwave}
\end{equation}
from \autoref{eq:planewave}. This is now used as a simple spherical wave. From
the Hertz dipole, it can be found that $A$ can be conveniently set to
$\frac{1}{r}=\frac{1}{\sqrt{x^2+y^2+z^2}}$ (see \cite{harrington}) and $k$ is
set to $2\,\pi$ for reasons of simplification.

Assume $n$ antennas are arranged as a circular array in the $xy$-plane at $z=0$.
Each antenna shall produce an electric field with
the amplitude given by \autoref{eq:simplesphericalwave}. If the near-field
couplings of these antennas are neglected, the far-fields can simply be
added together.
For the $i$-th antenna
\begin{equation}
E_i = \frac{\exp\left(-\j\,2\,\pi\sqrt{\left(x-x_i\right)^2+\left(y-y_i\right)^2+z^2}
+ \j\,\varphi_i \right)}{\sqrt{x^2+y^2+z^2}}  \label{eq:simplesphericalwave}
\end{equation}
yields the complex amplitude.   Note the
additionally introduced phase shift $\varphi_i$ which is needed to take the feed
phase shift into account.

The $x$- and $y$-positions of the individual antennas are found with
\begin{equation}
x_i = \varrho \cos \frac{2\,\pi\,i}{n}
\end{equation}
and
\begin{equation}
y_i = \varrho \sin \frac{2\,\pi\,i}{n}
\end{equation}
where $\varrho$ is the circular array radius. With
\begin{equation}
\varphi_i = \frac{2\,\pi\,i}{l}
\end{equation}
the phase shift associated with the OAM state $l$ is found.

The total field produced by all antennas together is then the superposition of
the individual fields:
\begin{equation}
E_{\mrm{tot}} = \sum\limits_{i=0}^{n-1} E_i
\end{equation}
This is then evaluated at a fixed $z$ value ($z = const.$).
\autoref{lst:matlabcode_oam} shows the implementation of this calculation in
\matlab{}.

The \matlab{} code was evaluated for different numbers of elements
and for different OAM states. \autoref{fig:matlabcalcu_array4_phase} shows a
comparison of the resulting figures for different OAM states of an array with
four elements.

\autoref{fig:matlabcalcu_array8_phase} (on page
\pageref{fig:matlabcalcu_array8_phase}) shows the different OAM states for an
eight element array.

\begin{figure}[ht]
  \centering
  \begin{gnuplot}[terminal = cairolatex, %
  terminaloptions = pdf input color solid]
  load 'plots/vorticeplots_4.gp'
  \end{gnuplot} %$
  \caption{Comparison of different OAM states of a four element array}
  \label{fig:matlabcalcu_array4_phase}
\end{figure}


\begin{lstlisting}[float, label = {lst:matlabcode_oam}, language=matlab, %
caption = {\matlab{} script to calculate the phase}]
clear all; close all force; clc

elem = 12; % number of elements in the array
l = 2; % oam state
r = 0.01; % array radius
k = 2*pi; % wavenumber
ang = (0:1:elem-1)*(2*pi/elem); % angular positions of the elements
xpos =  r*cos(ang); % x positions
ypos =  r*sin(ang); % y positions
phi = l*ang; % phase shift

% anonymous function to calculate the amplitude for one antenna
En = @(x, y, z, n) ...
    exp(-1i*k*sqrt((x-xpos(n)).^2 + (y-ypos(n)).^2 + z.^2) + 1i*phi(n));

% evaluate the total field at this z value
z = 10000;

[x, y] = meshgrid(linspace(-10, 10, 1000), linspace(-10, 10, 1000));
Etot = 0;

% add all fields together
for n = 1:length(ang)
    Etot = Etot + En(x, y, z, n);
end


figure(1);
pcolor(x, y, angle(Etot));
shading interp
axis square
\end{lstlisting}

\begin{figure}[ht]
  \centering
  \begin{gnuplot}[terminal = cairolatex, %
  terminaloptions = pdf input color solid]
  load 'plots/vorticeplots_8.gp'
  \end{gnuplot} %$
  \caption{Comparison of different OAM states of a eight element array}
  \label{fig:matlabcalcu_array8_phase}
\end{figure}