  We demonstrate an optical method to engineer optical Schr\"odinger cat states
(SCSs) of large size beta ranging from beta=2 to beta=3 with high fidelity
close to 0.99. Our approach uses the alpha-representation of the SCSs in
infinite Hilbert space with base in terms of displaced number states
characterized by the displacement amplitude alpha. An arbitrary
alpha-representation of SCSs enables manipulation of the amplitudes in wider
ranges of parameters, greatly expanding the possibilities for generation of
desired nonclassical states. The optical scheme we consider is quite universal
for implementation of the conditioned states close to SCSs with use of
linear-optics elements and detectors projecting unitarily transformed input
states onto the target one. Different states (e.g., number state or coherent
state or superposed state) are selected as the input to the optical scheme. In
particular, an input small-size Schroodinger kitten state can give rise to an
output large-size SCS.
