  Spin-orbit-coupled Bose-Einstein condensates (SOBECs) exhibit two new phases
of matter, now known as the stripe and plane-wave phases. When two interacting
spin components of a SOBEC spatially overlap, density modulations with
periodicity given by the spin-orbit coupling strength appear. In equilibrium,
these components fully overlap in the miscible stripe phase, and overlap only
in a domain wall in the immiscible plane-wave phase. Here we probe the density
modulation present in any overlapping region with optical Bragg scattering, and
observe the sudden drop of Bragg scattering as the overlapping region shrinks.
Using an atomic analogue of the Talbot effect, we demonstrate the existence of
long-range coherence between the different spin components in the stripe phase
and surprisingly even in the phase-separated plane-wave phase.
