 Quarter-wave matching layer for a quantum potential step

ID: quarter-wave-matching-layer-for-a-quantum-potential-step

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 = π /2

makes the two interface reflection amplitudes cancel exactly when

k_{1}^{2} = k_{0} k_{2}

. Equivalently

E - V_{1} = 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^{2 i k_{1} a}) / (1 + r_{01} r_{12} e^{2 i k_{1} a})

and

r_{ij} = (k_{i} - k_{j}) / (k_{i} + k_{j})

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