  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 the
magnetic field components perpendicular to the particle trajectory.
Equivalently, we express $P_{\gamma \to a}$ as the Fourier transform of the
magnetic field autocorrelation function, and establish a dictionary between
properties of the real-space magnetic field and the energy-dependent conversion
probability. For axions more massive than the plasma frequency,
($m_a>\omega_{\rm pl}$), we use this formalism to analytically solve the
problem of perturbative axion-photon mixing in a general magnetic field. In the
general case where $\omega_{\rm pl}/m_a$ varies arbitrarily along the
trajectory, we show that a naive application of the standard formalism for
'resonant' conversion can give highly inaccurate results, and that a careful
calculation generically gives non-resonant contributions at least as large as
the resonant contribution. Furthermore, we demonstrate how techniques based on
the Fast Fourier Transform provide a new, highly efficient numerical method for
calculating axion-photon mixing. We briefly discuss magnetic field modelling in
galaxy clusters in the light of our results and argue, in particular, that a
recently proposed 'regular' model used for studying axion-photon mixing
(specifically applied to the Perseus cluster) is inconsistent with
observations. Our formalism suggest new methods to search for imprints of
axions, and will be important for spectrographs with percent level sensitivity,
which includes existing X-ray observations by Chandra as well as the upcoming
Athena mission.
