  The second order optical response of centrosymmetric materials manifests
itself mostly at their surface, being strongly suppressed in their bulk.
However, the overall surface response is also suppressed in nanoparticles with
a centrosymmetric geometry subjected to homogeneous fields. Nevertheless,
nanoparticles with a noncentrosymmetric geometry do exhibit second order
optical properties. We develop an analytical theory to investigate the second
order optical response of a noncentrosymmetric thin cylinder with a slightly
deformed cross-section made up of a centrosymmetric material subjected to two
monochromatic fields. We calculate the linear and nonlinear near fields
perturbatively using the extent of the deformation away from a circular
cross-section as the perturbation parameter. We obtain expressions for the
quadratic hyperpolarizabilities in terms of the linear response evaluated at
the three frequencies involved. We analyze the spectral features of the
nonlinear response functions and explore their resonant structure for a model
dielectric cylinder. Furthermore, we evaluate the second order radiated fields,
the radiation patterns and efficiency of the different quadratic processes. We
obtain a strong competition between electric dipolar, magnetic dipolar and
electric quadrupolar contributions even for very small deformations.
