  We study GHZ-type and W-type three-mode entangled coherent states. Both the
types of entangled coherent states violate Mermin's version of the Bell
inequality with threshold photon detection (i.e., without photon counting).
Such an experiment can be performed using linear optics elements and threshold
detectors with significant Bell violations for GHZ-type entangled coherent
states. However, to demonstrate Bell-type inequality violations for W-type
entangled coherent states, additional nonlinear interactions are needed. We
also propose an optical scheme to generate W-type entangled coherent states in
free-traveling optical fields. The required resources for the generation are a
single-photon source, a coherent state source, beam splitters, phase shifters,
photodetectors, and Kerr nonlinearities. Our scheme does not necessarily
require strong Kerr nonlinear interactions, i.e., weak nonlinearities can be
used for the generation of the W-type entangled coherent states. Furthermore,
it is also robust against inefficiencies of the single-photon source and the
photon detectors.
