Possible higher-order paths through a five-level atom. Three possible paths are shown for the Ly-α transition through the intermediate level n = 3 expressed by γ₃. The green lime color highlights the different intermediate levels. Red arrows indicate the paths from the lower level n = 1 to the upper level n = 2. Blue arrows indicate the paths from the upper to the lower level. Upper panel: first-order paths indicated by ω₁ with $q_1^{31,2}=p_{31}(1-p_{54}{p}_{45})$ and $q_1^{32,1}=p_{32}(1-p_{54}{p}_{45})$. Middle panel: second-order paths indicated by ω₂ with $q_2^{31,2}=p_{34}{p}_{41}$ and $q_2^{32,1}=p_{34}{p}_{42}$. Bottom panel: third-order paths indicated by ω₃ with $q_3^{31,2}=p_{34}{p}_{45}{p}_{51}$ and $q_3^{32,1}=p_{34}{p}_{45}{p}_{52}$.