Figure 1: (Top) A small deformation of the lattice planes in a perfect crystal has a wave-guiding effect on an x-ray wave packet, effectively translating it by several mm through the crystal ($\mathbf{\text{G}}$ is a reciprocal lattice vector). (Bottom) A more detailed picture of how the curved lattice planes translate the x-ray “$\alpha$ mode” that propagates through the crystal. The purple lines indicate the lattice planes ($\mathbf{\text{u}}$ denotes the displacement as a function of space) and the blue lines denote the phase of the propagating $\alpha$ mode. As the $\alpha$ mode curves, the amplitudes of its constituent ${\mathbf{\text{A}}}_0$ and ${\mathbf{\text{A}}}_{\mathbf{\text{G}}}$ waves changes [6], with the amplitude of ${\mathbf{\text{A}}}_0$ increasing with respect to ${\mathbf{\text{A}}}_{\mathbf{\text{G}}}$. Since the Poynting vector is parallel to ${\left|{\mathbf{\text{A}}}_0\right|}^2{\mathbf{\text{k}}}_0+{\left|{\mathbf{\text{A}}}_{\mathbf{\text{G}}}\right|}^2({\mathbf{\text{k}}}_0+\mathbf{\text{G}})$, the exiting wave is mainly along ${\mathbf{\text{k}}}_0$.