  We study for the first time the effect of the geometry of quantum wire
networks on their nonlinear optical properties and show that for some
geometries, the first hyperpolarizability is largely enhanced and the second
hyperpolarizability is always negative or zero. We use a one-electron model
with tight transverse confinement. In the limit of infinite transverse
confinement, the transverse wavefunctions drop out of the
hyperpolarizabilities, but their residual effects are essential to include in
the sum rules. The effects of geometry are manifested in the projections of the
transition moments of each wire segment onto the 2-D lab frame. Numerical
optimization of the geometry of a loop leads to hyperpolarizabilities that
rival the best chromophores. We suggest that a combination of geometry and
quantum-confinement effects can lead to systems with ultralarge nonlinear
response.
