We present novel theory of effective realization of large-size optical Schrödinger cat states, which play an important role for quantum communication and quantum computation in the optical domain using laser sources. The treatment is based on the α-representation in infinite Hilbert space which is the decomposition of an arbitrary quantum state in terms of displaced number states characterized by the displacement amplitude α. We find analytical form of the α-representation for both even and odd Schrödinger cat states which is essential for their generation schemes. Two schemes are proposed for generating even/odd Schrödinger cat states of large size |β| (|β| ≥ 2) with high fidelity F (F ≈ 0.99). One scheme relies on an initially offline prepared two-mode entangled state with a fixed total photon number, while the other scheme uses separable photon Fock states as the input. In both schemes, generation of the desired states is heralded by the corresponding measurement outcomes. Conditions for obtaining states useful for quantum information processing are established and success probabilities for their generation are evaluated.