  Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic
rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia
set. We prove that $J$ always has positive Fourier dimension. The case where
$J$ is included in a circle follows from a recent work of Sahlsten and Stevens,
see arXiv:2009.01703. In the case where $J$ is not included in a circle, we
prove that a large family of probability measures supported on $J$ exhibit
polynomial Fourier decay: our result applies in particular to the measure of
maximal entropy and to the conformal measure.
