  We conduct systematic studies of the optical characteristics of plasmonic
nanoparticles that exhibit C2v symmetry. We analyze three distinct geometric
configurations: an L-type shape, a crescent, and a split-ring resonator.
Optical properties are examined using the FDTD method. It is demonstrated that
all three shapes exhibit two prominent plasmon bands associated with the two
axes of symmetry. This is in addition to a wide range of resonances observed at
high frequencies corresponding to quadrupole modes and peaks due to sharp
corners. Next, to facilitate nonlinear analysis, we employ a semiclassical
hydrodynamic model where the electron pressure term is explicitly accounted
for. Employing this model enables us to rigorously examine the second-order
angular resolved nonlinear optical response of these nanoparticles in each of
the three configurations. For CW pumping, we explore properties of the SHG.
Polarization and angle-resolved SHG spectra are obtained, revealing strong
dependence on the nanoparticle geometry and incident wave polarization. For
pulsed excitations, we discuss the phenomenon of broadband THz generation
induced by the DFG. It is shown that the THz emission spectra exhibit unique
features attributed to the plasmonic resonances and symmetry of the
nanoparticles. The polarization of the generated THz waves is also examined,
revealing interesting patterns tied to the nanoparticle geometry. To gain
deeper insight, we propose a simple analytical theory that agrees very well
with the numerical experiments. An expression for the far-field THz intensity
is derived in terms of the incident pulse parameters and the nonlinear response
tensor of the nanoparticle. The results presented in this work offer new
insights into the linear and nonlinear optical properties of nanoparticles with
C2v symmetry.
