expansion of an up (down) domain. The domain wall speed in a given direction can then be extracted from the ratio of the DW displacement and the pulse duration. DW displacements in the $\pm x$ directions were measured for a fixed value of the $H_{z}$ field, for several values of the in-plane field between -300 mT and +300 mT. In order to correct the residual $H_{z}$ component that may arise from a misalignment of the in-plane electromagnet, measurements were taken for both down and up domains.

Figure \[fig:images\] shows the differential images recorded in four positions of the wedge sample (called samples (A) to (D) from now on) corresponding to increasing values of the Gd thickness (from 2 to 5nm) for $H_{z}$ field pulses of the order of $10$mT and an in-plane field of $+200$mT. Without in-plane field, the propagation of the DWs is isotropic and the domains are round. Similar to previously reported experiments, the $H_{x}$ field breaks the rotational symmetry and the propagation becomes asymmetric in the $\pm x$ directions. Note that the sign and the amplitude of the speed asymmetry depend on the sample composition. Indeed, in sample (A) the down/up DWs move faster than the up/down 