A simple property of the Fourier transform is that if $f$ is real and even then its Fourier transform $\hat f$ is real, while if $f$ is real and odd then $\hat f$ is imaginary. This can be extended to the Fourier transform of a measure, saying that if $\mu$ is even ($\mu(E)=\mu(-E)$) then its Fourier transform is real. If we consider signed measure we can also say that if $\mu$ is odd then its Fourier transform is imaginary.
My question is: is there a continuous Radon non-negative (non trivial) measure whose Fourier transform lies on a line $y=ax$ in the complex plane (with $a\neq 0$)?
Of course, if we relax (some of) the hypotheses on the measure then we can get trivial examples. For example, since $\widehat{e^{i\alpha} f}= e^{i\alpha} \hat{f}$, we can "rotate" the Fourier transform of $f$. Therefore, for every line in the complex plane there is a complex function whose Fourier transform lies on that line. Similarly I can consider a complex measure and get an analogous result. Moreover if you take a delta then its Fourier transform is a point, and it lies on a line for trivial reasons, so I want the measure to be continuous.
Loosely speaking, is it true that if a measure (as above) is supported on $I\subset \mathbb{R}^ +$ then its Fourier transform "rotates" around the origin?