  Spin waves have the potential to be used as a new platform for data transfer
and processing as they can reach wavelengths in the nanometer range and
frequencies in the terahertz range. To realize a spin-wave device, it is
essential to be able to manipulate the amplitude as well as the phase of spin
waves. Several theoretical and recently also experimental works have shown that
the spin-wave phase can be manipulated by the transmission through a domain
wall (DW). Here, we study propagation of spin waves through a DW by means of
micro-focused Brillouin light scattering microscopy ($\mu$BLS). The acquired 2D
spin-wave intensity maps reveal that spin-wave transmission through a N\'eel DW
is influenced by a topologically enforced circular Bloch line in the DW center
and that the propagation regime depends on the spin-wave frequency. In the
first regime, two spin-wave beams propagating around the circular Bloch line
are formed, whereas in the second regime, spin waves propagate in a single
central beam through the circular Bloch line. Phase-resolved $\mu$BLS
measurements reveal a phase shift upon transmission through the domain wall for
both regimes. Micromagnetic modelling of the transmitted spin waves unveils a
distortion of their phase fronts which needs to be taken into account when
interpreting the measurements and designing potential devices. Moreover, we
show, by means of micromagnetic simulations, that an external magnetic field
can be used to move the circular Bloch line within the DW and to manipulate
spin-wave propagation.
