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We study the weak mixing of photons and relativistic axion-like particles(axions) in plasmas with background magnetic fields, ${\bf B}$. We show that,to leading order in the axion-photon coupling, the conversion probability,$P_{\gamma \to a}$, is given by the one-dimensional power spectrum of themagnetic field components perpendicular to the particle trajectory.Equivalently, we express $P_{\gamma \to a}$ as the Fourier transform of themagnetic field autocorrelation function, and establish a dictionary betweenproperties of the real-space magnetic field and the energy-dependent conversionprobability. For axions more massive than the plasma frequency,($m_a>\omega_{\rm pl}$), we use this formalism to analytically solve theproblem of perturbative axion-photon mixing in a general magnetic field. In thegeneral case where $\omega_{\rm pl}/m_a$ varies arbitrarily along thetrajectory, we show that a naive application of the standard formalism for'resonant' conversion can give highly inaccurate results, and that a carefulcalculation generically gives non-resonant contributions at least as large asthe resonant contribution. Furthermore, we demonstrate how techniques based onthe Fast Fourier Transform provide a new, highly efficient numerical method forcalculating axion-photon mixing. We briefly discuss magnetic field modelling ingalaxy clusters in the light of our results and argue, in particular, that arecently proposed 'regular' model used for studying axion-photon mixing(specifically applied to the Perseus cluster) is inconsistent withobservations. Our formalism suggest new methods to search for imprints ofaxions, and will be important for spectrographs with percent level sensitivity,which includes existing X-ray observations by Chandra as well as the upcomingAthena mission.

The Fourier formalism for relativistic axion-photon conversion, with astrophysical applications