  We propose two novel schemes to generate the two-dimensional 2xN and 3xN
cluster states by using a chain of (two-level) Rydberg atoms in the framework
of cavity QED. These schemes work in a completely deterministic way and are
based on the resonant interaction of the atoms in a chain with a bimodal cavity
that supports two independent modes of the photon field. We demonstrate that a
2xN cluster state can be generated efficiently with only a single bimodal
cavity, while two such cavities are needed in order to construct a 3xN cluster
state. It is shown, moreover, how these schemes can be extended towards the
generation of MxN two-dimensional cluster states.
