We describe the emergence of topological singularities in periodic media within the Ginzburg-Landau model and the core-radius approach. The energy functionals of both models are denoted by E_ε,δ, where ε represent the coherence length (in the Ginzburg-Landau model) or the core-radius size (in the core-radius approach) and δ denotes the periodicity scale. We carry out the Γ-convergence analysis of E_ε,δ as ε→ 0 and δ=δ_ε→ 0 in the |logε| scaling regime, showing that the Γ-limit consists in the

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... ergy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter (upon extraction of subsequences) λ=min{1,lim_ε→0|logδ_ε||logε|}, we show that in a sense we always have a separation-of-scale effect: at scales less than ε^λ we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than ε^λ the concentration process takes place "after" homogenization.