Quarter-wave matching layer for a quantum potential step ($k_{1}a=\pi /2,k_1^2=k_0{k}_2$)

For a constant-mass one-dimensional particle propagating through three regions of potentials $0,V_1,V_0$, let $k_j$ be the real positive wavevectors. A layer with $k_{1}a=\pi /2$ makes the two interface reflection amplitudes cancel exactly when $k_1^2=k_0{k}_2$. Equivalently $E-V_1=\sqrt{E(E-V_0)}$. For an increasing intermediate step this requires $E>V_0$ and $V_0/2<V_1<V_0$. The result follows from $r=(r_{01}+r_{12}{e}^{2i{k}_{1}a})/(1+r_{01}{r}_{12}{e}^{2i{k}_{1}a})$ and $r_{ij}=(k_i-k_j)/(k_i+k_j)$.