Past exam of the optics and electrodynamics course of the University of Cambridge / 2021 / Section A / A2

The transmitting antenna is matched to its input transmission line, so its real terminal electrical resistance is $Z_i$. The identical receiving antenna has the same electrical impedance at the operating frequency. Its available received power for a matched load is the incident energy flux times its antenna effective area:The real receiving load is $Z_t$, so its voltage-wave reflection coefficient relative to $Z_i$ is ${\Gamma }_L=(Z_t-Z_i)/(Z_t+Z_i)$. The fraction of available power delivered to the output line isConsequently, neglecting path losses, mismatch of polarization of an electromagnetic wave and line losses,This is the Friis transmission equation with a receiving impedance-mismatch factor. For fixed $P_i,G_a,d,Z_i,Z_t$, it scales as ${\lambda }^2$. That scaling is conditional on holding antenna gain fixed: a fixed physical aperture can instead have $G_a\propto {\lambda }^{-2}$, and any wavelength dependence of the impedances also enters the expression. The matched-output check $Z_t=Z_i$ gives $P_t=P_i\mathcal{F}$.

Distinguish reflection at the input terminal, reflection at the receiving load, and a returned signal that traverses the free-space link twice. The input antenna is initially matched, so its direct terminal reflection is zero.

In the usual weak-coupling, matched-port two-port network model for the antenna pair, let the forward and reverse wave amplitude transfer coefficients be $t_{21}$ and $t_{12}$. Electromagnetic reciprocity gives equal power transfer factors $|t_{21}{|}^2=|t_{12}{|}^2=\mathcal{F}$, where $\mathcal{F}=(G_a\lambda /(4\pi d){)}^2$ from part (a). The receiving-load reflection returns wave amplitude $t_{12}{\Gamma }_L{t}_{21}$ to the original input port. The load-mismatch contribution to the returned power is thereforeFor fixed gains and impedances this returned power scales as ${\lambda }^4$: it contains two link traversals. By contrast, the reflected power at the receiving load before that return traversal is $P_i\mathcal{F}|{\Gamma }_L{|}^2\propto {\lambda }^2$. Calling that local reflected power the power already back in the input line would omit one propagation loss.

The original wording does not specify an antenna scattering model. The boxed result is the load-dependent echo in an ideal matched-port link, with structural scattering and residual input coupling neglected. The actual total input return can include coherent scattering from the receiving antenna even when ${\Gamma }_L=0$; gains and effective areas alone do not specify that background. If “reflected back” means only the direct reflection at the stated matched transmitting antenna, its value is $0$. These meanings should not be conflated; the ${\lambda }^4$ result requires the two-pass echo interpretation and its stated model.