## 2 Background and Pseudo-Doppler Principles
### Application to OAM Radio Waves in the Far Field  
It is instructive to review the salient features of radio waves possessing various orders of OAM, which will be denoted by integers \(\pm k\). Such radio waves are generated by imposing a phase shift of \(k2\pi\) radians for every revolution of the observation point around the beam axis, giving it a helical phase front. This is not to be confused with polarization, which can be of any type. In RF applications, this can be relatively easily achieved using a uniform circular array of \(K\) identical antenna elements, each one fed by a current that is shifted in phase from that of its neighbor (in one direction) by \(k2\pi/K\) radians and with the same amplitude. Negative phase shifts generate OAM modes with helical phase fronts winding in the opposite sense around the beam axis, up to order \(K\)/2-1. A common method of creating multiple OAM beam excitations of the same circular array of antenna elements is to connect the \(K\) elements to the \(K\) output ports of a modified Butler matrix, and the \(K\) input ports of that Butler Matrix to \(K\) transmitters in the same RF band, with each modulated by a different stream of independent data symbols. The Butler matrix must be modified so as to  
Figure 1: Geometry for Doppler-based direction-finding. possess one port which gives rise to a zero-th order OAM mode; otherwise the electrical phases at the elements do not progress through an integer number of cycles so the phase fronts would not form continuous spirals. When plotted in three dimensions, the beam patterns appear conical for all non-zero orders of OAM, as depicted in Figure 2, where color was used to denote the electrical phase at a fixed time, modulo-\(2\pi\) radians. It is instructive to note that the phase (color) patterns rotate around the beam axis at the RF rate in time, i.e., one revolution per cycle of the radio frequency. Therefore, \(k\)-phase fronts (of a given color) pass a point on the cone of the \(k\)-th OAM beam in the tangential direction, per period of the RF carrier wave. Equivalently, at any given point in time, an electrical phase gradient of \(k2\pi\)/(\(2\pi R\)) radians per meter exists along the circular locus (also the beam footprint) around the axis of the conical beam of the \(k\)-th OAM mode. Figure 2: Far-field beam patterns of OAM modes \(k=0\) (top), 1, 2, 3, 4. In Figure 2, the antenna elements in the \(x\)-\(y\) plane numbered \(K=16\) and were omnidirectional, with the beam axes being in the \(z\) (vertical) direction. With the help of Figure 3, visualize the circular locus of the antenna element in Figure 1 as coinciding with the peak of a conical beam of OAM order \(k\), whose source is a circular array in the far field on the \(z\)-axis. Because the elevation angle of the source is now \(\varphi=0\), the only phase variation along the locus of the antenna element at a given point in time is due to that of the OAM beam. The shaded circle denotes the footprint area of the \(k\)-th OAM beam. Taking the phase at the \(x\)-axis as the reference phase of the moving antenna element, its phase at position \(\theta_{n}\) is therefore given simply by  
\[\varphi_{n}=k\theta_{n} \tag{3}\]  
at a given point in time. As the antenna moves around the circular locus in the \(x\)-\(y\) plane with uniform velocity \(v\), its angular position changes linearly with time, consequently causing its electrical phase to vary linearly with time according to (3) as  
\[\frac{d\varphi_{n}}{dt}=\frac{kd\theta_{n}}{dt} \tag{4}\]  
This can be related to a kind of \"transverse\" Doppler frequency shift because with \(R\) being constant, (4) can be written also as  
\[\frac{d\varphi_{n}}{dt}=\frac{kd\theta_{n}}{dt}=\frac{kv_{n}}{R}=2\pi f_{n, Doppler} \tag{5}\]  
Figure 3: Receiving antenna moving through OAM beam.