Stable surface solitons in truncated complex potentials

We show that surface solitons in the one-dimensional nonlinear Schr?dinger equation with truncated complex periodic potential can be stabilized by linear homogeneous losses, which are necessary to balance gain in the near-surface channel arising from the imaginary part of potential. Such solitons become stable attractors when the strength of homogeneous losses acquires values from a limited interval and they exist in focusing and defocusing media. The domains of stability of the surface solitons shrink with an increase in the amplitude of the imaginary part of complex potential.