  The polarization of Cosmic Microwave Background (CMB) photons is rotated as
they pass through (ultralight-) axion string loops. Studying this birefringence
can reveal valuable information about the axion-photon coupling and the
structure of the string network. We develop an approximate analytic formalism
and identify a kernel function that can be used to calculate the two-point
correlation function for CMB birefringence induced by an arbitrary axion string
network. Using this formalism, we evaluate the birefringence signal for some
simple loop distributions (including scaling and network collapse). We find
that the angular correlation function has a characteristic angular scale set by
$\theta_\mathrm{min}$, which corresponds to the angular extent of the loops at
the time of recombination. This results in a peak in the birefringence power
spectrum around $\ell_p \sim 1/\theta_\mathrm{min}$. An additional scale,
controlled by the axion's mass, is introduced if the network collapses before
today.
