Dipole-axis ambiguity from three intensity measurements

The angular distribution of electric dipole radiation measures squared projections of the dipole axis, so a few intensity measurements need not determine that axis or the total radiated power uniquely. For a real unit dipole axis $\mathbf{d}$, the radiation-zone formula is $I(\mathbf{n})=3P[1-(\mathbf{d}\cdot \mathbf{n}{)}^2]/(8\pi R^2)$. Equal intensities along $\hat{\mathbf{z}}$, $(\hat{\mathbf{x}}+\hat{\mathbf{z}})/\sqrt{2}$ and $(\hat{\mathbf{y}}+\hat{\mathbf{z}})/\sqrt{2}$ permit four axes, up to reversal:Three generally different radiated powers are compatible with the same intensity data. Additional directional measurements or polarization information can remove the ambiguity.