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Perelomov coherent states for equally spaced, infinite homogeneous waveguidearrays with Euclidean E(2) symmetry are defined, and a new resolution of theidentity is obtained. The key point to construct this novel resolution of theidentity is the fact that coherent states satisfy the Helmholtz equation (incoherent states labels), and thus every coherent state belongs to aone-parameter family uniquely determined by the Cauchy initial data of thecoherent state in a one-dimensional Cauchy set. For this reason we call\textit{Cauchy coherent} states to these initial data. The novel, non-localresolution of the identity in terms of Cauchy coherent states is provided usingframe theory. It is also shown that Perelomov coherent states for the EuclieanE(2) group have a simple and natural physical realization in these waveguidearrays.

Coherent States for infinite homogeneous waveguide arrays: Cauchy coherent states for $E(2)$