Past exam of the optics and electrodynamics course of the University of Cambridge / 2014 / Paper 1 / 1 / b / Solution

For the oblique coupling of parallel Hertzian dipoles, a lossless Hertzian dipole has directional gain $G(\theta )=\frac{3}{2}{\sin }^2\theta$. The separation is $r=\sqrt{2}d$, and both transmitting and receiving patterns have ${\sin }^2\theta =1/2$, so $G_t=G_r=3/4$. Reciprocity gives matched receiving effective area $A_e={\lambda }^2{G}_r/(4\pi )$. The emitted polarization matches the transverse projection of the receiving dipole axis, so there is no additional polarization mismatch. ConsequentlyBoth the off-axis transmitting pattern and the off-axis receiving aperture matter. This assumes the stated far field, conjugate matching and negligible dissipative antenna losses.