Phase maps for pupil plane apodization. Upper panel: coefficients (a, b) used to compute the phase apodizer as Ψ_α = aH₁ + bH₂. The solid lines show the optimised coefficient used in the final definition of the apodizer, i.e. the solution of Eq. (53). The dashed lines show the initial guess calculated from Eq. (51). The simulation has been made for K band observations (k₀ = 2π/2.2 µm) and the shape of the VLT pupil. Lower panels : the apodizer function, Ψ_α (u), for a target at an angular separation of 100 mas (left panel) and 200 mas (right panel).