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The spatial extension and complexity of the eigenfunctions of an openfinite-size two-dimensional (2D) random system are systematically studied for arandom collection of systems ranging from weakly scattering to localized. Theeigenfunctions are obtained by introducing gain in the medium and pumping justabove threshold. All lasing modes are found to correspond to quasimodes of thepassive system, for all regimes of propagation. We demonstrate the existence ofmultipeaked quasimodes or necklace states in 2D at the transition fromlocalized to diffusive, resulting from the coupling of localized states.

Complexity of 2D random laser modes at the transition from weak scattering to Anderson localization