  Understanding the magnetic and ferroelectric ordering of magnetoelectric
multiferroic materials at the nanoscale necessitates a versatile imaging method
with high spatial resolution. Here, soft X-ray ptychography is employed to
simultaneously image the ferroelectric and antiferromagnetic domains in an 80
nm thin freestanding film of the room-temperature multiferroic BiFeO$_3$ (BFO).
The antiferromagnetic spin cycloid of period 64 nm is resolved by
reconstructing the corresponding resonant elastic X-ray scattering in real
space and visualized together with mosaic-like ferroelectric domains in a
linear dichroic contrast image at the Fe L$_3$ edge. The measurements reveal a
near perfect coupling between the antiferromagnetic and ferroelectric ordering
by which the propagation direction of the spin cycloid is locked orthogonally
to the ferroelectric polarization. In addition, the study evinces both a
preference for in-plane propagation of the spin cycloid and changes of the
ferroelectric polarization by 71{\deg} between multiferroic domains in the
epitaxial strain-free, freestanding BFO film. The results provide a direct
visualization of the strong magnetoelectric coupling in BFO and of its fine
multiferroic domain structure, emphasizing the potential of ptychographic
imaging for the study of multiferroics and non-collinear magnetic materials
with soft X-rays.
