Two dipole antennas and their radiation patternThe simplest form of an antenna array are two short dipole antennas. In this problem you will learn how a phase difference between the currents of both antennas affects their combined radiation pattern.

Problem Statement

What is the combined radiation characteristic for two dipole antennas?
Two dipoles with different phase - what will be the outcome?

A single short dipole antenna with length \(L\ll\lambda\) shall obey a current\[\begin{eqnarray*}  I\left(z,t\right)&=&\begin{cases}
I_{0}e^{-\mathrm{i}\omega_{0}t}\left(1-2\frac{\left|z\right|}{L}\right) & \left|z\right|\leq L/2\\
0 & \mathrm{else}
\end{cases}\ .  \end{eqnarray*}\]Calculate the dipole moment \(\mathbf{p}\left(t\right)\) of such a current and its radiated power.

Now, suppose that two of these antennas are placed along the \(z\) axis with a separation \(d\). Assume that there is an intrinsic phase difference between both currents, say \(I_{0,2}=I_{0,1}\exp\left(-\mathrm{i}\Delta\phi_{21}^{0}\right)\). Then, what is the angular distribution of the radiated power \(dP/d\Omega\) in this case? Discuss the qualitative behavior of \(dP/d\Omega\) for \(\Delta\phi_{21}^{0}=0\) and \(\pi/2\) with both \(d=\lambda\) and \(\lambda/2\).

Background: Radiation of a Dipole

The time-averaged radiated power of a dipole with dipole moment \(\mathbf{p}\) oscillating at some angular frequency \(\omega_{0}\) is given by\[\begin{eqnarray*}  P&=&\frac{\omega_{0}^{4}}{12\pi\varepsilon_{0}c^{3}}\left\langle \mathbf{p}^{2}\right\rangle \ .  \end{eqnarray*}\]This relation can be calculated from the time-averaged Poynting vector of such a source in the farfield,\[\begin{eqnarray*}  \left\langle \mathbf{S}\right\rangle &=&\frac{\omega_{0}^{4}}{32\pi^{2}\varepsilon_{0}c^{3}}\frac{\left\langle \mathbf{p}^{2}\right\rangle \sin^{2}\theta}{r^{2}}\mathbf{e}_{r}\equiv\frac{1}{r^{2}}\frac{dP}{d\Omega}\mathbf{e}_{r}\ ,  \end{eqnarray*}\]where \(\theta\in\left[0,\pi\right]\) is the angle between \(\mathbf{p}\) and a vector at the position of (thought) measurement. If \(\mathbf{p}\) is along the \(z\) axis, this angle is the usual polar angle in spherical coordinates. The quantity \(dP/d\Omega\) is usually termed the angular distribution of the radiated power and goes characteristically with \(\sin^{2}\theta\) for a dipole. Note that as usual \(d\Omega=\sin\theta d\theta d\varphi\).

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